深度学习中的数学知识

必备数学知识

深度学习中的数学知识_第1张图片


我们了解了第一课-深度学习的基本理论以后,可以开始不断的去深入了解背后的原理是什么。为什么图片能被计算机读取?为什么我们可以用CNN对成千上万中图片进行分类,这背后的原理是什么?在了解原理之前,先给大家补点数学知识,因为无论是深度学习还是机器学习,背后都是有一些数学原理和公式推导的,所以掌握必备的数学知识必不可少,下面会给大家简单科普下常用的数学知识有哪些~

数学基础知识

数据科学需要一定的数学基础,但仅仅做应用的话,如果时间不多,不用学太深,了解基本公式即可,遇到问题再查吧。

下面是常见的一些数学基础概念,建议大家收藏后再仔细阅读,遇到不懂的概念可以直接在这里查~

高等数学

1.导数定义:

导数和微分的概念

f′(x0)=lim⁡Δx→0 f(x0+Δx)−f(x0)Δxf'({{x}_{0}})=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}f′(x0​)=Δx→0lim​Δxf(x0​+Δx)−f(x0​)​ (1)

或者:

f′(x0)=lim⁡x→x0 f(x)−f(x0)x−x0f'({{x}_{0}})=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}}f′(x0​)=x→x0​lim​x−x0​f(x)−f(x0​)​ (2)

2.左右导数导数的几何意义和物理意义

函数f(x)f(x)f(x)在x0x_0x0​处的左、右导数分别定义为:

左导数:f′−(x0)=lim⁡Δx→0− f(x0+Δx)−f(x0)Δx=lim⁡x→x0− f(x)−f(x0)x−x0,(x=x0+Δx){{{f}'}_{-}}({{x}_{0}})=\underset{\Delta x\to {{0}^{-}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{-}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}},(x={{x}_{0}}+\Delta x)f′−​(x0​)=Δx→0−lim​Δxf(x0​+Δx)−f(x0​)​=x→x0−​lim​x−x0​f(x)−f(x0​)​,(x=x0​+Δx)

右导数:f′+(x0)=lim⁡Δx→0+ f(x0+Δx)−f(x0)Δx=lim⁡x→x0+ f(x)−f(x0)x−x0{{{f}'}_{+}}({{x}_{0}})=\underset{\Delta x\to {{0}^{+}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{+}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}}f′+​(x0​)=Δx→0+lim​Δxf(x0​+Δx)−f(x0​)​=x→x0+​lim​x−x0​f(x)−f(x0​)​

3.函数的可导性与连续性之间的关系

Th1: 函数f(x)f(x)f(x)在x0x_0x0​处可微⇔f(x)\Leftrightarrow f(x)⇔f(x)在x0x_0x0​处可导

Th2: 若函数在点x0x_0x0​处可导,则y=f(x)y=f(x)y=f(x)在点x0x_0x0​处连续,反之则不成立。即函数连续不一定可导。

Th3: f′(x0){f}'({{x}_{0}})f′(x0​)存在⇔f′−(x0)=f′+(x0)\Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}})⇔f′−​(x0​)=f′+​(x0​)

4.平面曲线的切线和法线

切线方程 : y−y0=f′(x0)(x−x0)y-{{y}_{0}}=f'({{x}_{0}})(x-{{x}_{0}})y−y0​=f′(x0​)(x−x0​) 法线方程:y−y0=−1f′(x0)(x−x0),f′(x0)≠0y-{{y}_{0}}=-\frac{1}{f'({{x}_{0}})}(x-{{x}_{0}}),f'({{x}_{0}})\ne 0y−y0​=−f′(x0​)1​(x−x0​),f′(x0​)=0

5.四则运算法则 设函数u=u(x),v=v(x)u=u(x),v=v(x)u=u(x),v=v(x)]在点xxx可导则 (1) (u±v)′=u′±v′(u\pm v{)}'={u}'\pm {v}'(u±v)′=u′±v′ d(u±v)=du±dvd(u\pm v)=du\pm dvd(u±v)=du±dv (2)(uv)′=uv′+vu′(uv{)}'=u{v}'+v{u}'(uv)′=uv′+vu′ d(uv)=udv+vdud(uv)=udv+vdud(uv)=udv+vdu (3) (uv)′=vu′−uv′v2(v≠0)(\frac{u}{v}{)}'=\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\ne 0)(vu​)′=v2vu′−uv′​(v=0) d(uv)=vdu−udvv2d(\frac{u}{v})=\frac{vdu-udv}{{{v}^{2}}}d(vu​)=v2vdu−udv​

6.基本导数与微分表 (1) y=cy=cy=c(常数) y′=0{y}'=0y′=0 dy=0dy=0dy=0 (2) y=xαy={{x}^{\alpha }}y=xα(α\alphaα为实数) y′=αxα−1{y}'=\alpha {{x}^{\alpha -1}}y′=αxα−1 dy=αxα−1dxdy=\alpha {{x}^{\alpha -1}}dxdy=αxα−1dx (3) y=axy={{a}^{x}}y=ax y′=axln⁡a{y}'={{a}^{x}}\ln ay′=axlna dy=axln⁡adxdy={{a}^{x}}\ln adxdy=axlnadx 特例: (ex)′=ex({{{e}}^{x}}{)}'={{{e}}^{x}}(ex)′=ex d(ex)=exdxd({{{e}}^{x}})={{{e}}^{x}}dxd(ex)=exdx

(4) y=log⁡axy={{\log }_{a}}xy=loga​x y′=1xln⁡a{y}'=\frac{1}{x\ln a}y′=xlna1​

dy=1xln⁡adxdy=\frac{1}{x\ln a}dxdy=xlna1​dx 特例:y=ln⁡xy=\ln xy=lnx (ln⁡x)′=1x(\ln x{)}'=\frac{1}{x}(lnx)′=x1​ d(ln⁡x)=1xdxd(\ln x)=\frac{1}{x}dxd(lnx)=x1​dx

(5) y=sin⁡xy=\sin xy=sinx

y′=cos⁡x{y}'=\cos xy′=cosx d(sin⁡x)=cos⁡xdxd(\sin x)=\cos xdxd(sinx)=cosxdx

(6) y=cos⁡xy=\cos xy=cosx

y′=−sin⁡x{y}'=-\sin xy′=−sinx d(cos⁡x)=−sin⁡xdxd(\cos x)=-\sin xdxd(cosx)=−sinxdx

(7) y=tan⁡xy=\tan xy=tanx

y′=1cos⁡2x=sec⁡2x{y}'=\frac{1}{{{\cos }^{2}}x}={{\sec }^{2}}xy′=cos2x1​=sec2x d(tan⁡x)=sec⁡2xdxd(\tan x)={{\sec }^{2}}xdxd(tanx)=sec2xdx (8) y=cot⁡xy=\cot xy=cotx y′=−1sin⁡2x=−csc⁡2x{y}'=-\frac{1}{{{\sin }^{2}}x}=-{{\csc }^{2}}xy′=−sin2x1​=−csc2x d(cot⁡x)=−csc⁡2xdxd(\cot x)=-{{\csc }^{2}}xdxd(cotx)=−csc2xdx (9) y=sec⁡xy=\sec xy=secx y′=sec⁡xtan⁡x{y}'=\sec x\tan xy′=secxtanx

d(sec⁡x)=sec⁡xtan⁡xdxd(\sec x)=\sec x\tan xdxd(secx)=secxtanxdx (10) y=csc⁡xy=\csc xy=cscx y′=−csc⁡xcot⁡x{y}'=-\csc x\cot xy′=−cscxcotx

d(csc⁡x)=−csc⁡xcot⁡xdxd(\csc x)=-\csc x\cot xdxd(cscx)=−cscxcotxdx (11) y=arcsin⁡xy=\arcsin xy=arcsinx

y′=11−x2{y}'=\frac{1}{\sqrt{1-{{x}^{2}}}}y′=1−x2​1​

d(arcsin⁡x)=11−x2dxd(\arcsin x)=\frac{1}{\sqrt{1-{{x}^{2}}}}dxd(arcsinx)=1−x2​1​dx (12) y=arccos⁡xy=\arccos xy=arccosx

y′=−11−x2{y}'=-\frac{1}{\sqrt{1-{{x}^{2}}}}y′=−1−x2​1​ d(arccos⁡x)=−11−x2dxd(\arccos x)=-\frac{1}{\sqrt{1-{{x}^{2}}}}dxd(arccosx)=−1−x2​1​dx

(13) y=arctan⁡xy=\arctan xy=arctanx

y′=11+x2{y}'=\frac{1}{1+{{x}^{2}}}y′=1+x21​ d(arctan⁡x)=11+x2dxd(\arctan x)=\frac{1}{1+{{x}^{2}}}dxd(arctanx)=1+x21​dx

(14) y=arc⁡cot⁡xy=\operatorname{arc}\cot xy=arccotx

y′=−11+x2{y}'=-\frac{1}{1+{{x}^{2}}}y′=−1+x21​

d(arc⁡cot⁡x)=−11+x2dxd(\operatorname{arc}\cot x)=-\frac{1}{1+{{x}^{2}}}dxd(arccotx)=−1+x21​dx (15) y=shxy=shxy=shx

y′=chx{y}'=chxy′=chx d(shx)=chxdxd(shx)=chxdxd(shx)=chxdx

(16) y=chxy=chxy=chx

y′=shx{y}'=shxy′=shx d(chx)=shxdxd(chx)=shxdxd(chx)=shxdx

7.复合函数,反函数,隐函数以及参数方程所确定的函数的微分法

(1) 反函数的运算法则: 设y=f(x)y=f(x)y=f(x)在点xxx的某邻域内单调连续,在点xxx处可导且f′(x)≠0{f}'(x)\ne 0f′(x)=0,则其反函数在点xxx所对应的yyy处可导,并且有dydx=1dxdy\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}dxdy​=dydx​1​ (2) 复合函数的运算法则:若 μ=φ(x)\mu =\varphi(x)μ=φ(x) 在点xxx可导,而y=f(μ)y=f(\mu)y=f(μ)在对应点μ\muμ(μ=φ(x)\mu =\varphi (x)μ=φ(x))可导,则复合函数y=f(φ(x))y=f(\varphi (x))y=f(φ(x))在点xxx可导,且y′=f′(μ)⋅φ′(x){y}'={f}'(\mu )\cdot {\varphi }'(x)y′=f′(μ)⋅φ′(x) (3) 隐函数导数dydx\frac{dy}{dx}dxdy​的求法一般有三种方法: 1)方程两边对xxx求导,要记住yyy是xxx的函数,则yyy的函数是xxx的复合函数.例如1y\frac{1}{y}y1​,y2{{y}^{2}}y2,lnyln ylny,ey{{{e}}^{y}}ey等均是xxx的复合函数. 对xxx求导应按复合函数连锁法则做. 2)公式法.由F(x,y)=0F(x,y)=0F(x,y)=0知 dydx=−F′x(x,y)F′y(x,y)\frac{dy}{dx}=-\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)}dxdy​=−F′y​(x,y)F′x​(x,y)​,其中,F′x(x,y){{{F}'}_{x}}(x,y)F′x​(x,y), F′y(x,y){{{F}'}_{y}}(x,y)F′y​(x,y)分别表示F(x,y)F(x,y)F(x,y)对xxx和yyy的偏导数 3)利用微分形式不变性

8.常用高阶导数公式

(1)(ax) (n)=axln⁡na(a>0)(ex) (n)=e x({{a}^{x}}){{\,}^{(n)}}={{a}^{x}}{{\ln }^{n}}a\quad (a>{0})\quad \quad ({{{e}}^{x}}){{\,}^{(n)}}={e}{{\,}^{x}}(ax)(n)=axlnna(a>0)(ex)(n)=ex (2)(sin⁡kx) (n)=knsin⁡(kx+n⋅π2)(\sin kx{)}{{\,}^{(n)}}={{k}^{n}}\sin (kx+n\cdot \frac{\pi }{{2}})(sinkx)(n)=knsin(kx+n⋅2π​) (3)(cos⁡kx) (n)=kncos⁡(kx+n⋅π2)(\cos kx{)}{{\,}^{(n)}}={{k}^{n}}\cos (kx+n\cdot \frac{\pi }{{2}})(coskx)(n)=kncos(kx+n⋅2π​) (4)(xm) (n)=m(m−1)⋯(m−n+1)xm−n({{x}^{m}}){{\,}^{(n)}}=m(m-1)\cdots (m-n+1){{x}^{m-n}}(xm)(n)=m(m−1)⋯(m−n+1)xm−n (5)(ln⁡x) (n)=(−1)(n−1)(n−1)!xn(\ln x){{\,}^{(n)}}={{(-{1})}^{(n-{1})}}\frac{(n-{1})!}{{{x}^{n}}}(lnx)(n)=(−1)(n−1)xn(n−1)!​ (6)莱布尼兹公式:若u(x) ,v(x)u(x)\,,v(x)u(x),v(x)均nnn阶可导,则 (uv)(n)=∑i=0ncniu(i)v(n−i){{(uv)}^{(n)}}=\sum\limits_{i={0}}^{n}{c_{n}^{i}{{u}^{(i)}}{{v}^{(n-i)}}}(uv)(n)=i=0∑n​cni​u(i)v(n−i),其中u(0)=u{{u}^{({0})}}=uu(0)=u,v(0)=v{{v}^{({0})}}=vv(0)=v

9.微分中值定理,泰勒公式

Th1:(费马定理)

若函数f(x)f(x)f(x)满足条件: (1)函数f(x)f(x)f(x)在x0{{x}_{0}}x0​的某邻域内有定义,并且在此邻域内恒有 f(x)≤f(x0)f(x)\le f({{x}_{0}})f(x)≤f(x0​)或f(x)≥f(x0)f(x)\ge f({{x}_{0}})f(x)≥f(x0​),

(2) f(x)f(x)f(x)在x0{{x}_{0}}x0​处可导,则有 f′(x0)=0{f}'({{x}_{0}})=0f′(x0​)=0

Th2:(罗尔定理)

设函数f(x)f(x)f(x)满足条件: (1)在闭区间[a,b][a,b][a,b]上连续;

(2)在(a,b)(a,b)(a,b)内可导;

(3)f(a)=f(b)f(a)=f(b)f(a)=f(b);

则在(a,b)(a,b)(a,b)内一存在个$\xi $,使 f′(ξ)=0{f}'(\xi )=0f′(ξ)=0 Th3: (拉格朗日中值定理)

设函数f(x)f(x)f(x)满足条件: (1)在[a,b][a,b][a,b]上连续;

(2)在(a,b)(a,b)(a,b)内可导;

则在(a,b)(a,b)(a,b)内一存在个$\xi $,使 f(b)−f(a)b−a=f′(ξ)\frac{f(b)-f(a)}{b-a}={f}'(\xi )b−af(b)−f(a)​=f′(ξ)

Th4: (柯西中值定理)

设函数f(x)f(x)f(x),g(x)g(x)g(x)满足条件: (1) 在[a,b][a,b][a,b]上连续;

(2) 在(a,b)(a,b)(a,b)内可导且f′(x){f}'(x)f′(x),g′(x){g}'(x)g′(x)均存在,且g′(x)≠0{g}'(x)\ne 0g′(x)=0

则在(a,b)(a,b)(a,b)内存在一个$\xi $,使 f(b)−f(a)g(b)−g(a)=f′(ξ)g′(ξ)\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{{f}'(\xi )}{{g}'(\xi )}g(b)−g(a)f(b)−f(a)​=g′(ξ)f′(ξ)​

10.洛必达法则 法则Ⅰ (00\frac{0}{0}00​型) 设函数f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x)满足条件: lim⁡x→x0 f(x)=0,lim⁡x→x0 g(x)=0\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=0x→x0​lim​f(x)=0,x→x0​lim​g(x)=0;

f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x)在x0{{x}_{0}}x0​的邻域内可导,(在x0{{x}_{0}}x0​处可除外)且g′(x)≠0{g}'\left( x \right)\ne 0g′(x)=0;

lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g′(x)f′(x)​存在(或$\infty $)。

则: lim⁡x→x0 f(x)g(x)=lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g(x)f(x)​=x→x0​lim​g′(x)f′(x)​。 法则I′{{I}'}I′ (00\frac{0}{0}00​型)设函数f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x)满足条件: lim⁡x→∞ f(x)=0,lim⁡x→∞ g(x)=0\underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to \infty }{\mathop{\lim }}\,g\left( x \right)=0x→∞lim​f(x)=0,x→∞lim​g(x)=0;

存在一个X>0X>0X>0,当∣x∣>X\left| x \right|>X∣x∣>X时,f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x)可导,且g′(x)≠0{g}'\left( x \right)\ne 0g′(x)=0;lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g′(x)f′(x)​存在(或$\infty $)。

则lim⁡x→x0 f(x)g(x)=lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g(x)f(x)​=x→x0​lim​g′(x)f′(x)​ 法则Ⅱ( ∞∞\frac{\infty }{\infty }∞∞​ 型) 设函数 f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x) 满足条件: lim⁡x→x0 f(x)=∞,lim⁡x→x0 g(x)=∞\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=\infty,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=\inftyx→x0​lim​f(x)=∞,x→x0​lim​g(x)=∞;
f(x),g(x)f\left( x \right),g\left( x \right)f(x),g(x) 在 x0{{x}_{0}}x0​ 的邻域内可导(在x0{{x}_{0}}x0​处可除外)且g′(x)≠0{g}'\left( x \right)\ne 0g′(x)=0;lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g′(x)f′(x)​ 存在(或$\infty )。则)。 则)。则lim⁡x→x0 f(x)g(x)=lim⁡x→x0 f′(x)g′(x)\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}x→x0​lim​g(x)f(x)​=x→x0​lim​g′(x)f′(x)​$ 同理法则II′{I{I}'}II′ ( ∞∞\frac{\infty }{\infty }∞∞​ 型)仿法则 I′{{I}'}I′ 可写出。

11.泰勒公式

设函数f(x)f(x)f(x)在点x0{{x}_{0}}x0​处的某邻域内具有n+1n+1n+1阶导数,则对该邻域内异于x0{{x}_{0}}x0​的任意点xxx,在x0{{x}_{0}}x0​与xxx之间至少存在 一个ξ\xiξ,使得: f(x)=f(x0)+f′(x0)(x−x0)+12!f′′(x0)(x−x0)2+⋯f(x)=f({{x}_{0}})+{f}'({{x}_{0}})(x-{{x}_{0}})+\frac{1}{2!}{f}''({{x}_{0}}){{(x-{{x}_{0}})}^{2}}+\cdotsf(x)=f(x0​)+f′(x0​)(x−x0​)+2!1​f′′(x0​)(x−x0​)2+⋯ +f(n)(x0)n!(x−x0)n+Rn(x)+\frac{{{f}^{(n)}}({{x}_{0}})}{n!}{{(x-{{x}_{0}})}^{n}}+{{R}_{n}}(x)+n!f(n)(x0​)​(x−x0​)n+Rn​(x) 其中 Rn(x)=f(n+1)(ξ)(n+1)!(x−x0)n+1{{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{(x-{{x}_{0}})}^{n+1}}Rn​(x)=(n+1)!f(n+1)(ξ)​(x−x0​)n+1称为f(x)f(x)f(x)在点x0{{x}_{0}}x0​处的nnn阶泰勒余项。

令x0=0{{x}_{0}}=0x0​=0,则nnn阶泰勒公式 f(x)=f(0)+f′(0)x+12!f′′(0)x2+⋯+f(n)(0)n!xn+Rn(x)f(x)=f(0)+{f}'(0)x+\frac{1}{2!}{f}''(0){{x}^{2}}+\cdots +\frac{{{f}^{(n)}}(0)}{n!}{{x}^{n}}+{{R}_{n}}(x)f(x)=f(0)+f′(0)x+2!1​f′′(0)x2+⋯+n!f(n)(0)​xn+Rn​(x)……(1) 其中 Rn(x)=f(n+1)(ξ)(n+1)!xn+1{{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{x}^{n+1}}Rn​(x)=(n+1)!f(n+1)(ξ)​xn+1,$\xi 在0与在0与在0与x$之间.(1)式称为麦克劳林公式

常用五种函数在x0=0{{x}_{0}}=0x0​=0处的泰勒公式

(1) ex=1+x+12!x2+⋯+1n!xn+xn+1(n+1)!eξ{{{e}}^{x}}=1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+\frac{{{x}^{n+1}}}{(n+1)!}{{e}^{\xi }}ex=1+x+2!1​x2+⋯+n!1​xn+(n+1)!xn+1​eξ

或 =1+x+12!x2+⋯+1n!xn+o(xn)=1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+o({{x}^{n}})=1+x+2!1​x2+⋯+n!1​xn+o(xn)

(2) sin⁡x=x−13!x3+⋯+xnn!sin⁡nπ2+xn+1(n+1)!sin⁡(ξ+n+12π)\sin x=x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\sin (\xi +\frac{n+1}{2}\pi )sinx=x−3!1​x3+⋯+n!xn​sin2nπ​+(n+1)!xn+1​sin(ξ+2n+1​π)

或 =x−13!x3+⋯+xnn!sin⁡nπ2+o(xn)=x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+o({{x}^{n}})=x−3!1​x3+⋯+n!xn​sin2nπ​+o(xn)

(3) cos⁡x=1−12!x2+⋯+xnn!cos⁡nπ2+xn+1(n+1)!cos⁡(ξ+n+12π)\cos x=1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\cos (\xi +\frac{n+1}{2}\pi )cosx=1−2!1​x2+⋯+n!xn​cos2nπ​+(n+1)!xn+1​cos(ξ+2n+1​π)

或 =1−12!x2+⋯+xnn!cos⁡nπ2+o(xn)=1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+o({{x}^{n}})=1−2!1​x2+⋯+n!xn​cos2nπ​+o(xn)

(4) ln⁡(1+x)=x−12x2+13x3−⋯+(−1)n−1xnn+(−1)nxn+1(n+1)(1+ξ)n+1\ln (1+x)=x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+\frac{{{(-1)}^{n}}{{x}^{n+1}}}{(n+1){{(1+\xi )}^{n+1}}}ln(1+x)=x−21​x2+31​x3−⋯+(−1)n−1nxn​+(n+1)(1+ξ)n+1(−1)nxn+1​

或 =x−12x2+13x3−⋯+(−1)n−1xnn+o(xn)=x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+o({{x}^{n}})=x−21​x2+31​x3−⋯+(−1)n−1nxn​+o(xn)

(5) (1+x)m=1+mx+m(m−1)2!x2+⋯+m(m−1)⋯(m−n+1)n!xn{{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}(1+x)m=1+mx+2!m(m−1)​x2+⋯+n!m(m−1)⋯(m−n+1)​xn +m(m−1)⋯(m−n+1)(n+1)!xn+1(1+ξ)m−n−1+\frac{m(m-1)\cdots (m-n+1)}{(n+1)!}{{x}^{n+1}}{{(1+\xi )}^{m-n-1}}+(n+1)!m(m−1)⋯(m−n+1)​xn+1(1+ξ)m−n−1

或(1+x)m=1+mx+m(m−1)2!x2+⋯{{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots(1+x)m=1+mx+2!m(m−1)​x2+⋯ ,+m(m−1)⋯(m−n+1)n!xn+o(xn)+\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}+o({{x}^{n}})+n!m(m−1)⋯(m−n+1)​xn+o(xn)

12.函数单调性的判断 Th1: 设函数f(x)f(x)f(x)在(a,b)(a,b)(a,b)区间内可导,如果对∀x∈(a,b)\forall x\in (a,b)∀x∈(a,b),都有f ′(x)>0f\,'(x)>0f′(x)>0(或f ′(x)<0f\,'(x)<0f′(x)<0),则函数f(x)f(x)f(x)在(a,b)(a,b)(a,b)内是单调增加的(或单调减少)

Th2: (取极值的必要条件)设函数f(x)f(x)f(x)在x0{{x}_{0}}x0​处可导,且在x0{{x}_{0}}x0​处取极值,则f ′(x0)=0f\,'({{x}_{0}})=0f′(x0​)=0。

Th3: (取极值的第一充分条件)设函数f(x)f(x)f(x)在x0{{x}_{0}}x0​的某一邻域内可微,且f ′(x0)=0f\,'({{x}_{0}})=0f′(x0​)=0(或f(x)f(x)f(x)在x0{{x}_{0}}x0​处连续,但f ′(x0)f\,'({{x}_{0}})f′(x0​)不存在。) (1)若当xxx经过x0{{x}_{0}}x0​时,f ′(x)f\,'(x)f′(x)由“+”变“-”,则f(x0)f({{x}_{0}})f(x0​)为极大值; (2)若当xxx经过x0{{x}_{0}}x0​时,f ′(x)f\,'(x)f′(x)由“-”变“+”,则f(x0)f({{x}_{0}})f(x0​)为极小值; (3)若f ′(x)f\,'(x)f′(x)经过x=x0x={{x}_{0}}x=x0​的两侧不变号,则f(x0)f({{x}_{0}})f(x0​)不是极值。

Th4: (取极值的第二充分条件)设f(x)f(x)f(x)在点x0{{x}_{0}}x0​处有f′′(x)≠0f''(x)\ne 0f′′(x)=0,且f ′(x0)=0f\,'({{x}_{0}})=0f′(x0​)=0,则 当f′ ′(x0)<0f'\,'({{x}_{0}})<0f′′(x0​)<0时,f(x0)f({{x}_{0}})f(x0​)为极大值; 当f′ ′(x0)>0f'\,'({{x}_{0}})>0f′′(x0​)>0时,f(x0)f({{x}_{0}})f(x0​)为极小值。 注:如果f′ ′(x0)<0f'\,'({{x}_{0}})<0f′′(x0​)<0,此方法失效。

13.渐近线的求法 (1)水平渐近线 若lim⁡x→+∞ f(x)=b\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=bx→+∞lim​f(x)=b,或lim⁡x→−∞ f(x)=b\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=bx→−∞lim​f(x)=b,则

y=by=by=b称为函数y=f(x)y=f(x)y=f(x)的水平渐近线。

(2)铅直渐近线 若$$\underset{x\to x_{0}^{-}}{\mathop{\lim }},f(x)=\infty$$,或$$\underset{x\to x_{0}^{+}}{\mathop{\lim }},f(x)=\infty$$,则

x=x0x={{x}_{0}}x=x0​称为y=f(x)y=f(x)y=f(x)的铅直渐近线。

(3)斜渐近线 若a=lim⁡x→∞ f(x)x,b=lim⁡x→∞ [f(x)−ax]a=\underset{x\to \infty }{\mathop{\lim }}\,\frac{f(x)}{x},\quad b=\underset{x\to \infty }{\mathop{\lim }}\,[f(x)-ax]a=x→∞lim​xf(x)​,b=x→∞lim​[f(x)−ax],则 y=ax+by=ax+by=ax+b称为y=f(x)y=f(x)y=f(x)的斜渐近线。

14.函数凹凸性的判断 Th1: (凹凸性的判别定理)若在I上f′′(x)<0f''(x)<0f′′(x)<0(或f′′(x)>0f''(x)>0f′′(x)>0),则f(x)f(x)f(x)在I上是凸的(或凹的)。

Th2: (拐点的判别定理1)若在x0{{x}_{0}}x0​处f′′(x)=0f''(x)=0f′′(x)=0,(或f′′(x)f''(x)f′′(x)不存在),当xxx变动经过x0{{x}_{0}}x0​时,f′′(x)f''(x)f′′(x)变号,则(x0,f(x0))({{x}_{0}},f({{x}_{0}}))(x0​,f(x0​))为拐点。

Th3: (拐点的判别定理2)设f(x)f(x)f(x)在x0{{x}_{0}}x0​点的某邻域内有三阶导数,且f′′(x)=0f''(x)=0f′′(x)=0,f′′′(x)≠0f'''(x)\ne 0f′′′(x)=0,则(x0,f(x0))({{x}_{0}},f({{x}_{0}}))(x0​,f(x0​))为拐点。

15.弧微分

dS=1+y′2dxdS=\sqrt{1+y{{'}^{2}}}dxdS=1+y′2​dx

16.曲率

曲线y=f(x)y=f(x)y=f(x)在点(x,y)(x,y)(x,y)处的曲率k=∣y′′∣(1+y′2)32k=\frac{\left| y'' \right|}{{{(1+y{{'}^{2}})}^{\tfrac{3}{2}}}}k=(1+y′2)23​∣y′′∣​。 对于参数方程\left\{ \begin{align} & x=\varphi (t) \\ & y=\psi (t) \\ \end{align}\right., k=∣φ′(t)ψ′′(t)−φ′′(t)ψ′(t)∣[φ′2(t)+ψ′2(t)]32k=\frac{\left| \varphi '(t)\psi ''(t)-\varphi ''(t)\psi '(t) \right|}{{{[\varphi {{'}^{2}}(t)+\psi {{'}^{2}}(t)]}^{\tfrac{3}{2}}}}k=[φ′2(t)+ψ′2(t)]23​∣φ′(t)ψ′′(t)−φ′′(t)ψ′(t)∣​。

17.曲率半径

曲线在点MMM处的曲率k(k≠0)k(k\ne 0)k(k=0)与曲线在点MMM处的曲率半径ρ\rhoρ有如下关系:ρ=1k\rho =\frac{1}{k}ρ=k1​。

线性代数

行列式

1.行列式按行(列)展开定理

(1) 设A=(aij)n×nA = ( a_{{ij}} )_{n \times n}A=(aij​)n×n​,则:ai1Aj1+ai2Aj2+⋯+ainAjn={∣A∣,i=j0,i≠ja_{i1}A_{j1} +a_{i2}A_{j2} + \cdots + a_{{in}}A_{{jn}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases}ai1​Aj1​+ai2​Aj2​+⋯+ain​Ajn​={∣A∣,i=j0,i=j​

或a1iA1j+a2iA2j+⋯+aniAnj={∣A∣,i=j0,i≠ja_{1i}A_{1j} + a_{2i}A_{2j} + \cdots + a_{{ni}}A_{{nj}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases}a1i​A1j​+a2i​A2j​+⋯+ani​Anj​={∣A∣,i=j0,i=j​即 AA∗=A∗A=∣A∣E,AA^{*} = A^{*}A = \left| A \right|E,AA∗=A∗A=∣A∣E,其中:A∗=(A11A12…A1nA21A22…A2n…………An1An2…Ann)=(Aji)=(Aij)TA^{*} = \begin{pmatrix} A_{11} & A_{12} & \ldots & A_{1n} \\ A_{21} & A_{22} & \ldots & A_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ A_{n1} & A_{n2} & \ldots & A_{{nn}} \\ \end{pmatrix} = (A_{{ji}}) = {(A_{{ij}})}^{T}A∗=⎝⎜⎜⎜⎛​A11​A21​…An1​​A12​A22​…An2​​…………​A1n​A2n​…Ann​​⎠⎟⎟⎟⎞​=(Aji​)=(Aij​)T

Dn=∣11…1x1x2…xn…………x1n−1x2n−1…xnn−1∣=∏1≤j

(2) 设A,BA,BA,B为nnn阶方阵,则∣AB∣=∣A∣∣B∣=∣B∣∣A∣=∣BA∣\left| {AB} \right| = \left| A \right|\left| B \right| = \left| B \right|\left| A \right| = \left| {BA} \right|∣AB∣=∣A∣∣B∣=∣B∣∣A∣=∣BA∣,但∣A±B∣=∣A∣±∣B∣\left| A \pm B \right| = \left| A \right| \pm \left| B \right|∣A±B∣=∣A∣±∣B∣不一定成立。

(3) ∣kA∣=kn∣A∣\left| {kA} \right| = k^{n}\left| A \right|∣kA∣=kn∣A∣,AAA为nnn阶方阵。

(4) 设AAA为nnn阶方阵,∣AT∣=∣A∣;∣A−1∣=∣A∣−1|A^{T}| = |A|;|A^{- 1}| = |A|^{- 1}∣AT∣=∣A∣;∣A−1∣=∣A∣−1(若AAA可逆),∣A∗∣=∣A∣n−1|A^{*}| = |A|^{n - 1}∣A∗∣=∣A∣n−1

n≥2n \geq 2n≥2

(5) ∣AOOB∣=∣ACOB∣=∣AOCB∣=∣A∣∣B∣\left| \begin{matrix} & {A\quad O} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad C} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad O} \\ & {C\quad B} \\ \end{matrix} \right| =| A||B|∣∣∣∣∣​​AOOB​∣∣∣∣∣​=∣∣∣∣∣​​ACOB​∣∣∣∣∣​=∣∣∣∣∣​​AOCB​∣∣∣∣∣​=∣A∣∣B∣ ,A,BA,BA,B为方阵,但∣OAm×mBn×nO∣=(−1)mn∣A∣∣B∣\left| \begin{matrix} {O} & A_{m \times m} \\ B_{n \times n} & { O} \\ \end{matrix} \right| = ({- 1)}^{{mn}}|A||B|∣∣∣∣∣​OBn×n​​Am×m​O​∣∣∣∣∣​=(−1)mn∣A∣∣B∣ 。

(6) 范德蒙行列式Dn=∣11…1x1x2…xn…………x1n−1x2n1…xnn−1∣=∏1≤j

设AAA是nnn阶方阵,λi(i=1,2⋯ ,n)\lambda_{i}(i = 1,2\cdots,n)λi​(i=1,2⋯,n)是AAA的nnn个特征值,则 ∣A∣=∏i=1nλi|A| = \prod_{i = 1}^{n}\lambda_{i}∣A∣=∏i=1n​λi​

矩阵

矩阵:m×nm \times nm×n个数aija_{{ij}}aij​排成mmm行nnn列的表格[a11a12⋯a1na21a22⋯a2n⋯⋯⋯⋯⋯am1am2⋯amn]\begin{bmatrix} a_{11}\quad a_{12}\quad\cdots\quad a_{1n} \\ a_{21}\quad a_{22}\quad\cdots\quad a_{2n} \\ \quad\cdots\cdots\cdots\cdots\cdots \\ a_{m1}\quad a_{m2}\quad\cdots\quad a_{{mn}} \\ \end{bmatrix}⎣⎢⎢⎢⎡​a11​a12​⋯a1n​a21​a22​⋯a2n​⋯⋯⋯⋯⋯am1​am2​⋯amn​​⎦⎥⎥⎥⎤​ 称为矩阵,简记为AAA,或者(aij)m×n\left( a_{{ij}} \right)_{m \times n}(aij​)m×n​ 。若m=nm = nm=n,则称AAA是nnn阶矩阵或nnn阶方阵。

矩阵的线性运算

1.矩阵的加法

设A=(aij),B=(bij)A = (a_{{ij}}),B = (b_{{ij}})A=(aij​),B=(bij​)是两个m×nm \times nm×n矩阵,则m×nm \times nm×n 矩阵C=cij)=aij+bijC = c_{{ij}}) = a_{{ij}} + b_{{ij}}C=cij​)=aij​+bij​称为矩阵AAA与BBB的和,记为A+B=CA + B = CA+B=C 。

2.矩阵的数乘

设A=(aij)A = (a_{{ij}})A=(aij​)是m×nm \times nm×n矩阵,kkk是一个常数,则m×nm \times nm×n矩阵(kaij)(ka_{{ij}})(kaij​)称为数kkk与矩阵AAA的数乘,记为kA{kA}kA。

3.矩阵的乘法

设A=(aij)A = (a_{{ij}})A=(aij​)是m×nm \times nm×n矩阵,B=(bij)B = (b_{{ij}})B=(bij​)是n×sn \times sn×s矩阵,那么m×sm \times sm×s矩阵C=(cij)C = (c_{{ij}})C=(cij​),其中cij=ai1b1j+ai2b2j+⋯+ainbnj=∑k=1naikbkjc_{{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{{in}}b_{{nj}} = \sum_{k =1}^{n}{a_{{ik}}b_{{kj}}}cij​=ai1​b1j​+ai2​b2j​+⋯+ain​bnj​=∑k=1n​aik​bkj​称为AB{AB}AB的乘积,记为C=ABC = ABC=AB 。

4. AT\mathbf{A}^{\mathbf{T}}ATA−1\mathbf{A}^{\mathbf{-1}}A−1A∗\mathbf{A}^{\mathbf{*}}A∗三者之间的关系

(1) (AT)T=A,(AB)T=BTAT,(kA)T=kAT,(A±B)T=AT±BT{(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \pm B)}^{T} = A^{T} \pm B^{T}(AT)T=A,(AB)T=BTAT,(kA)T=kAT,(A±B)T=AT±BT

(2) (A−1)−1=A,(AB)−1=B−1A−1,(kA)−1=1kA−1,\left( A^{- 1} \right)^{- 1} = A,\left( {AB} \right)^{- 1} = B^{- 1}A^{- 1},\left( {kA} \right)^{- 1} = \frac{1}{k}A^{- 1},(A−1)−1=A,(AB)−1=B−1A−1,(kA)−1=k1​A−1,

但 (A±B)−1=A−1±B−1{(A \pm B)}^{- 1} = A^{- 1} \pm B^{- 1}(A±B)−1=A−1±B−1不一定成立。

(3) (A∗)∗=∣A∣n−2 A  (n≥3)\left( A^{*} \right)^{*} = |A|^{n - 2}\ A\ \ (n \geq 3)(A∗)∗=∣A∣n−2 A  (n≥3),(AB)∗=B∗A∗,\left({AB} \right)^{*} = B^{*}A^{*},(AB)∗=B∗A∗, (kA)∗=kn−1A∗  (n≥2)\left( {kA} \right)^{*} = k^{n -1}A^{*}{\ \ }\left( n \geq 2 \right)(kA)∗=kn−1A∗  (n≥2)

但(A±B)∗=A∗±B∗\left( A \pm B \right)^{*} = A^{*} \pm B^{*}(A±B)∗=A∗±B∗不一定成立。

(4) (A−1)T=(AT)−1, (A−1)∗=(AA∗)−1,(A∗)T=(AT)∗{(A^{- 1})}^{T} = {(A^{T})}^{- 1},\ \left( A^{- 1} \right)^{*} ={(AA^{*})}^{- 1},{(A^{*})}^{T} = \left( A^{T} \right)^{*}(A−1)T=(AT)−1, (A−1)∗=(AA∗)−1,(A∗)T=(AT)∗

5.有关A∗\mathbf{A}^{\mathbf{*}}A∗的结论

(1) AA∗=A∗A=∣A∣EAA^{*} = A^{*}A = |A|EAA∗=A∗A=∣A∣E

(2) ∣A∗∣=∣A∣n−1 (n≥2),    (kA)∗=kn−1A∗,  (A∗)∗=∣A∣n−2A(n≥3)|A^{*}| = |A|^{n - 1}\ (n \geq 2),\ \ \ \ {(kA)}^{*} = k^{n -1}A^{*},{{\ \ }\left( A^{*} \right)}^{*} = |A|^{n - 2}A(n \geq 3)∣A∗∣=∣A∣n−1 (n≥2),    (kA)∗=kn−1A∗,  (A∗)∗=∣A∣n−2A(n≥3)

(3) 若AAA可逆,则A∗=∣A∣A−1,(A∗)∗=1∣A∣AA^{*} = |A|A^{- 1},{(A^{*})}^{*} = \frac{1}{|A|}AA∗=∣A∣A−1,(A∗)∗=∣A∣1​A

(4) 若AAA为nnn阶方阵,则:

r(A∗)={n,r(A)=n1,r(A)=n−10,r(A)

6.有关A−1\mathbf{A}^{\mathbf{- 1}}A−1的结论

AAA可逆⇔AB=E;⇔∣A∣≠0;⇔r(A)=n;\Leftrightarrow AB = E; \Leftrightarrow |A| \neq 0; \Leftrightarrow r(A) = n;⇔AB=E;⇔∣A∣=0;⇔r(A)=n;

⇔A\Leftrightarrow A⇔A可以表示为初等矩阵的乘积;⇔A;⇔Ax=0\Leftrightarrow A;\Leftrightarrow Ax = 0⇔A;⇔Ax=0。

7.有关矩阵秩的结论

(1) 秩r(A)r(A)r(A)=行秩=列秩;

(2) r(Am×n)≤min⁡(m,n);r(A_{m \times n}) \leq \min(m,n);r(Am×n​)≤min(m,n);

(3) A≠0⇒r(A)≥1A \neq 0 \Rightarrow r(A) \geq 1A=0⇒r(A)≥1;

(4) r(A±B)≤r(A)+r(B);r(A \pm B) \leq r(A) + r(B);r(A±B)≤r(A)+r(B);

(5) 初等变换不改变矩阵的秩

(6) r(A)+r(B)−n≤r(AB)≤min⁡(r(A),r(B)),r(A) + r(B) - n \leq r(AB) \leq \min(r(A),r(B)),r(A)+r(B)−n≤r(AB)≤min(r(A),r(B)),特别若AB=OAB = OAB=O 则:r(A)+r(B)≤nr(A) + r(B) \leq nr(A)+r(B)≤n

(7) 若A−1A^{- 1}A−1存在⇒r(AB)=r(B);\Rightarrow r(AB) = r(B);⇒r(AB)=r(B); 若B−1B^{- 1}B−1存在 ⇒r(AB)=r(A);\Rightarrow r(AB) = r(A);⇒r(AB)=r(A);

若r(Am×n)=n⇒r(AB)=r(B);r(A_{m \times n}) = n \Rightarrow r(AB) = r(B);r(Am×n​)=n⇒r(AB)=r(B); 若r(Am×s)=n⇒r(AB)=r(A)r(A_{m \times s}) = n\Rightarrow r(AB) = r\left( A \right)r(Am×s​)=n⇒r(AB)=r(A)。

(8) r(Am×s)=n⇔Ax=0r(A_{m \times s}) = n \Leftrightarrow Ax = 0r(Am×s​)=n⇔Ax=0只有零解

8.分块求逆公式

(AOOB)−1=(A−1OOB−1)\begin{pmatrix} A & O \\ O & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{-1} & O \\ O & B^{- 1} \\ \end{pmatrix}(AO​OB​)−1=(A−1O​OB−1​); (ACOB)−1=(A−1−A−1CB−1OB−1)\begin{pmatrix} A & C \\ O & B \\\end{pmatrix}^{- 1} = \begin{pmatrix} A^{- 1}& - A^{- 1}CB^{- 1} \\ O & B^{- 1} \\ \end{pmatrix}(AO​CB​)−1=(A−1O​−A−1CB−1B−1​);

(AOCB)−1=(A−1O−B−1CA−1B−1)\begin{pmatrix} A & O \\ C & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{- 1}&{O} \\ - B^{- 1}CA^{- 1} & B^{- 1} \\\end{pmatrix}(AC​OB​)−1=(A−1−B−1CA−1​OB−1​); (OABO)−1=(OB−1A−1O)\begin{pmatrix} O & A \\ B & O \\ \end{pmatrix}^{- 1} =\begin{pmatrix} O & B^{- 1} \\ A^{- 1} & O \\ \end{pmatrix}(OB​AO​)−1=(OA−1​B−1O​)

这里AAA,BBB均为可逆方阵。

向量

1.有关向量组的线性表示

(1)α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性相关⇔\Leftrightarrow⇔至少有一个向量可以用其余向量线性表示。

(2)α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性无关,α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​,β\betaβ线性相关⇔β\Leftrightarrow \beta⇔β可以由α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​唯一线性表示。

(3) β\betaβ可以由α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性表示 ⇔r(α1,α2,⋯ ,αs)=r(α1,α2,⋯ ,αs,β)\Leftrightarrow r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}) =r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta)⇔r(α1​,α2​,⋯,αs​)=r(α1​,α2​,⋯,αs​,β) 。

2.有关向量组的线性相关性

(1)部分相关,整体相关;整体无关,部分无关.

(2) ① nnn个nnn维向量 α1,α2⋯αn\alpha_{1},\alpha_{2}\cdots\alpha_{n}α1​,α2​⋯αn​线性无关⇔∣[α1α2⋯αn]∣≠0\Leftrightarrow \left|\left\lbrack \alpha_{1}\alpha_{2}\cdots\alpha_{n} \right\rbrack \right| \neq0⇔∣[α1​α2​⋯αn​]∣=0, nnn个nnn维向量α1,α2⋯αn\alpha_{1},\alpha_{2}\cdots\alpha_{n}α1​,α2​⋯αn​线性相关 ⇔∣[α1,α2,⋯ ,αn]∣=0\Leftrightarrow |\lbrack\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\rbrack| = 0⇔∣[α1​,α2​,⋯,αn​]∣=0 。

② n+1n + 1n+1个nnn维向量线性相关。

③ 若α1,α2⋯αS\alpha_{1},\alpha_{2}\cdots\alpha_{S}α1​,α2​⋯αS​线性无关,则添加分量后仍线性无关;或一组向量线性相关,去掉某些分量后仍线性相关。

3.有关向量组的线性表示

(1) α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性相关⇔\Leftrightarrow⇔至少有一个向量可以用其余向量线性表示。

(2) α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性无关,α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​,β\betaβ线性相关⇔β\Leftrightarrow\beta⇔β 可以由α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​唯一线性表示。

(3) β\betaβ可以由α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性表示 ⇔r(α1,α2,⋯ ,αs)=r(α1,α2,⋯ ,αs,β)\Leftrightarrow r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}) =r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta)⇔r(α1​,α2​,⋯,αs​)=r(α1​,α2​,⋯,αs​,β)

4.向量组的秩与矩阵的秩之间的关系

设r(Am×n)=rr(A_{m \times n}) =rr(Am×n​)=r,则AAA的秩r(A)r(A)r(A)与AAA的行列向量组的线性相关性关系为:

(1) 若r(Am×n)=r=mr(A_{m \times n}) = r = mr(Am×n​)=r=m,则AAA的行向量组线性无关。

(2) 若r(Am×n)=r

(3) 若r(Am×n)=r=nr(A_{m \times n}) = r = nr(Am×n​)=r=n,则AAA的列向量组线性无关。

(4) 若r(Am×n)=r

5.n\mathbf{n}n维向量空间的基变换公式及过渡矩阵

若α1,α2,⋯ ,αn\alpha_{1},\alpha_{2},\cdots,\alpha_{n}α1​,α2​,⋯,αn​与β1,β2,⋯ ,βn\beta_{1},\beta_{2},\cdots,\beta_{n}β1​,β2​,⋯,βn​是向量空间VVV的两组基,则基变换公式为:

(β1,β2,⋯ ,βn)=(α1,α2,⋯ ,αn)[c11c12⋯c1nc21c22⋯c2n⋯⋯⋯⋯cn1cn2⋯cnn]=(α1,α2,⋯ ,αn)C(\beta_{1},\beta_{2},\cdots,\beta_{n}) = (\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\begin{bmatrix} c_{11}& c_{12}& \cdots & c_{1n} \\ c_{21}& c_{22}&\cdots & c_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ c_{n1}& c_{n2} & \cdots & c_{{nn}} \\\end{bmatrix} = (\alpha_{1},\alpha_{2},\cdots,\alpha_{n})C(β1​,β2​,⋯,βn​)=(α1​,α2​,⋯,αn​)⎣⎢⎢⎢⎡​c11​c21​⋯cn1​​c12​c22​⋯cn2​​⋯⋯⋯⋯​c1n​c2n​⋯cnn​​⎦⎥⎥⎥⎤​=(α1​,α2​,⋯,αn​)C

其中CCC是可逆矩阵,称为由基α1,α2,⋯ ,αn\alpha_{1},\alpha_{2},\cdots,\alpha_{n}α1​,α2​,⋯,αn​到基β1,β2,⋯ ,βn\beta_{1},\beta_{2},\cdots,\beta_{n}β1​,β2​,⋯,βn​的过渡矩阵。

6.坐标变换公式

若向量γ\gammaγ在基α1,α2,⋯ ,αn\alpha_{1},\alpha_{2},\cdots,\alpha_{n}α1​,α2​,⋯,αn​与基β1,β2,⋯ ,βn\beta_{1},\beta_{2},\cdots,\beta_{n}β1​,β2​,⋯,βn​的坐标分别是 X=(x1,x2,⋯ ,xn)TX = {(x_{1},x_{2},\cdots,x_{n})}^{T}X=(x1​,x2​,⋯,xn​)T,

Y=(y1,y2,⋯ ,yn)TY = \left( y_{1},y_{2},\cdots,y_{n} \right)^{T}Y=(y1​,y2​,⋯,yn​)T 即: γ=x1α1+x2α2+⋯+xnαn=y1β1+y2β2+⋯+ynβn\gamma =x_{1}\alpha_{1} + x_{2}\alpha_{2} + \cdots + x_{n}\alpha_{n} = y_{1}\beta_{1} +y_{2}\beta_{2} + \cdots + y_{n}\beta_{n}γ=x1​α1​+x2​α2​+⋯+xn​αn​=y1​β1​+y2​β2​+⋯+yn​βn​,则向量坐标变换公式为X=CYX = CYX=CY 或Y=C−1XY = C^{- 1}XY=C−1X,其中CCC是从基α1,α2,⋯ ,αn\alpha_{1},\alpha_{2},\cdots,\alpha_{n}α1​,α2​,⋯,αn​到基β1,β2,⋯ ,βn\beta_{1},\beta_{2},\cdots,\beta_{n}β1​,β2​,⋯,βn​的过渡矩阵。

7.向量的内积

(α,β)=a1b1+a2b2+⋯+anbn=αTβ=βTα(\alpha,\beta) = a_{1}b_{1} + a_{2}b_{2} + \cdots + a_{n}b_{n} = \alpha^{T}\beta = \beta^{T}\alpha(α,β)=a1​b1​+a2​b2​+⋯+an​bn​=αTβ=βTα

8.Schmidt正交化

若α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​线性无关,则可构造β1,β2,⋯ ,βs\beta_{1},\beta_{2},\cdots,\beta_{s}β1​,β2​,⋯,βs​使其两两正交,且βi\beta_{i}βi​仅是α1,α2,⋯ ,αi\alpha_{1},\alpha_{2},\cdots,\alpha_{i}α1​,α2​,⋯,αi​的线性组合(i=1,2,⋯ ,n)(i= 1,2,\cdots,n)(i=1,2,⋯,n),再把βi\beta_{i}βi​单位化,记γi=βi∣βi∣\gamma_{i} =\frac{\beta_{i}}{\left| \beta_{i}\right|}γi​=∣βi​∣βi​​,则γ1,γ2,⋯ ,γi\gamma_{1},\gamma_{2},\cdots,\gamma_{i}γ1​,γ2​,⋯,γi​是规范正交向量组。其中 β1=α1\beta_{1} = \alpha_{1}β1​=α1​, β2=α2−(α2,β1)(β1,β1)β1\beta_{2} = \alpha_{2} -\frac{(\alpha_{2},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1}β2​=α2​−(β1​,β1​)(α2​,β1​)​β1​ , β3=α3−(α3,β1)(β1,β1)β1−(α3,β2)(β2,β2)β2\beta_{3} =\alpha_{3} - \frac{(\alpha_{3},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} -\frac{(\alpha_{3},\beta_{2})}{(\beta_{2},\beta_{2})}\beta_{2}β3​=α3​−(β1​,β1​)(α3​,β1​)​β1​−(β2​,β2​)(α3​,β2​)​β2​ ,

............

βs=αs−(αs,β1)(β1,β1)β1−(αs,β2)(β2,β2)β2−⋯−(αs,βs−1)(βs−1,βs−1)βs−1\beta_{s} = \alpha_{s} - \frac{(\alpha_{s},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} - \frac{(\alpha_{s},\beta_{2})}{(\beta_{2},\beta_{2})}\beta_{2} - \cdots - \frac{(\alpha_{s},\beta_{s - 1})}{(\beta_{s - 1},\beta_{s - 1})}\beta_{s - 1}βs​=αs​−(β1​,β1​)(αs​,β1​)​β1​−(β2​,β2​)(αs​,β2​)​β2​−⋯−(βs−1​,βs−1​)(αs​,βs−1​)​βs−1​

9.正交基及规范正交基

向量空间一组基中的向量如果两两正交,就称为正交基;若正交基中每个向量都是单位向量,就称其为规范正交基。

线性方程组

1.克莱姆法则

线性方程组{a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋯⋯⋯⋯⋯⋯⋯⋯⋯an1x1+an2x2+⋯+annxn=bn\begin{cases} a_{11}x_{1} + a_{12}x_{2} + \cdots +a_{1n}x_{n} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} =b_{2} \\ \quad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \\ a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{{nn}}x_{n} = b_{n} \\ \end{cases}⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧​a11​x1​+a12​x2​+⋯+a1n​xn​=b1​a21​x1​+a22​x2​+⋯+a2n​xn​=b2​⋯⋯⋯⋯⋯⋯⋯⋯⋯an1​x1​+an2​x2​+⋯+ann​xn​=bn​​,如果系数行列式D=∣A∣≠0D = \left| A \right| \neq 0D=∣A∣=0,则方程组有唯一解,x1=D1D,x2=D2D,⋯ ,xn=DnDx_{1} = \frac{D_{1}}{D},x_{2} = \frac{D_{2}}{D},\cdots,x_{n} =\frac{D_{n}}{D}x1​=DD1​​,x2​=DD2​​,⋯,xn​=DDn​​,其中DjD_{j}Dj​是把DDD中第jjj列元素换成方程组右端的常数列所得的行列式。

2. nnn阶矩阵AAA可逆⇔Ax=0\Leftrightarrow Ax = 0⇔Ax=0只有零解。⇔∀b,Ax=b\Leftrightarrow\forall b,Ax = b⇔∀b,Ax=b总有唯一解,一般地,r(Am×n)=n⇔Ax=0r(A_{m \times n}) = n \Leftrightarrow Ax= 0r(Am×n​)=n⇔Ax=0只有零解。

3.非奇次线性方程组有解的充分必要条件,线性方程组解的性质和解的结构

(1) 设AAA为m×nm \times nm×n矩阵,若r(Am×n)=mr(A_{m \times n}) = mr(Am×n​)=m,则对Ax=bAx =bAx=b而言必有r(A)=r(A⋮b)=mr(A) = r(A \vdots b) = mr(A)=r(A⋮b)=m,从而Ax=bAx = bAx=b有解。

(2) 设x1,x2,⋯xsx_{1},x_{2},\cdots x_{s}x1​,x2​,⋯xs​为Ax=bAx = bAx=b的解,则k1x1+k2x2⋯+ksxsk_{1}x_{1} + k_{2}x_{2}\cdots + k_{s}x_{s}k1​x1​+k2​x2​⋯+ks​xs​当k1+k2+⋯+ks=1k_{1} + k_{2} + \cdots + k_{s} = 1k1​+k2​+⋯+ks​=1时仍为Ax=bAx =bAx=b的解;但当k1+k2+⋯+ks=0k_{1} + k_{2} + \cdots + k_{s} = 0k1​+k2​+⋯+ks​=0时,则为Ax=0Ax =0Ax=0的解。特别x1+x22\frac{x_{1} + x_{2}}{2}2x1​+x2​​为Ax=bAx = bAx=b的解;2x3−(x1+x2)2x_{3} - (x_{1} +x_{2})2x3​−(x1​+x2​)为Ax=0Ax = 0Ax=0的解。

(3) 非齐次线性方程组Ax=b{Ax} = bAx=b无解⇔r(A)+1=r(A‾)⇔b\Leftrightarrow r(A) + 1 =r(\overline{A}) \Leftrightarrow b⇔r(A)+1=r(A)⇔b不能由AAA的列向量α1,α2,⋯ ,αn\alpha_{1},\alpha_{2},\cdots,\alpha_{n}α1​,α2​,⋯,αn​线性表示。

4.奇次线性方程组的基础解系和通解,解空间,非奇次线性方程组的通解

(1) 齐次方程组Ax=0{Ax} = 0Ax=0恒有解(必有零解)。当有非零解时,由于解向量的任意线性组合仍是该齐次方程组的解向量,因此Ax=0{Ax}= 0Ax=0的全体解向量构成一个向量空间,称为该方程组的解空间,解空间的维数是n−r(A)n - r(A)n−r(A),解空间的一组基称为齐次方程组的基础解系。

(2) η1,η2,⋯ ,ηt\eta_{1},\eta_{2},\cdots,\eta_{t}η1​,η2​,⋯,ηt​是Ax=0{Ax} = 0Ax=0的基础解系,即:

  1. η1,η2,⋯ ,ηt\eta_{1},\eta_{2},\cdots,\eta_{t}η1​,η2​,⋯,ηt​是Ax=0{Ax} = 0Ax=0的解;

  2. η1,η2,⋯ ,ηt\eta_{1},\eta_{2},\cdots,\eta_{t}η1​,η2​,⋯,ηt​线性无关;

  3. Ax=0{Ax} = 0Ax=0的任一解都可以由η1,η2,⋯ ,ηt\eta_{1},\eta_{2},\cdots,\eta_{t}η1​,η2​,⋯,ηt​线性表出. k1η1+k2η2+⋯+ktηtk_{1}\eta_{1} + k_{2}\eta_{2} + \cdots + k_{t}\eta_{t}k1​η1​+k2​η2​+⋯+kt​ηt​是Ax=0{Ax} = 0Ax=0的通解,其中k1,k2,⋯ ,ktk_{1},k_{2},\cdots,k_{t}k1​,k2​,⋯,kt​是任意常数。

矩阵的特征值和特征向量

1.矩阵的特征值和特征向量的概念及性质

(1) 设λ\lambdaλ是AAA的一个特征值,则 kA,aA+bE,A2,Am,f(A),AT,A−1,A∗{kA},{aA} + {bE},A^{2},A^{m},f(A),A^{T},A^{- 1},A^{*}kA,aA+bE,A2,Am,f(A),AT,A−1,A∗有一个特征值分别为 kλ,aλ+b,λ2,λm,f(λ),λ,λ−1,∣A∣λ,{kλ},{aλ} + b,\lambda^{2},\lambda^{m},f(\lambda),\lambda,\lambda^{- 1},\frac{|A|}{\lambda},kλ,aλ+b,λ2,λm,f(λ),λ,λ−1,λ∣A∣​,且对应特征向量相同(ATA^{T}AT 例外)。

(2)若λ1,λ2,⋯ ,λn\lambda_{1},\lambda_{2},\cdots,\lambda_{n}λ1​,λ2​,⋯,λn​为AAA的nnn个特征值,则∑i=1nλi=∑i=1naii,∏i=1nλi=∣A∣\sum_{i= 1}^{n}\lambda_{i} = \sum_{i = 1}^{n}a_{{ii}},\prod_{i = 1}^{n}\lambda_{i}= |A|∑i=1n​λi​=∑i=1n​aii​,∏i=1n​λi​=∣A∣ ,从而∣A∣≠0⇔A|A| \neq 0 \Leftrightarrow A∣A∣=0⇔A没有特征值。

(3)设λ1,λ2,⋯ ,λs\lambda_{1},\lambda_{2},\cdots,\lambda_{s}λ1​,λ2​,⋯,λs​为AAA的sss个特征值,对应特征向量为α1,α2,⋯ ,αs\alpha_{1},\alpha_{2},\cdots,\alpha_{s}α1​,α2​,⋯,αs​,

若: α=k1α1+k2α2+⋯+ksαs\alpha = k_{1}\alpha_{1} + k_{2}\alpha_{2} + \cdots + k_{s}\alpha_{s}α=k1​α1​+k2​α2​+⋯+ks​αs​ ,

则: Anα=k1Anα1+k2Anα2+⋯+ksAnαs=k1λ1nα1+k2λ2nα2+⋯ksλsnαsA^{n}\alpha = k_{1}A^{n}\alpha_{1} + k_{2}A^{n}\alpha_{2} + \cdots +k_{s}A^{n}\alpha_{s} = k_{1}\lambda_{1}^{n}\alpha_{1} +k_{2}\lambda_{2}^{n}\alpha_{2} + \cdots k_{s}\lambda_{s}^{n}\alpha_{s}Anα=k1​Anα1​+k2​Anα2​+⋯+ks​Anαs​=k1​λ1n​α1​+k2​λ2n​α2​+⋯ks​λsn​αs​ 。

2.相似变换、相似矩阵的概念及性质

(1) 若A∼BA \sim BA∼B,则

  1. AT∼BT,A−1∼B−1,,A∗∼B∗A^{T} \sim B^{T},A^{- 1} \sim B^{- 1},,A^{*} \sim B^{*}AT∼BT,A−1∼B−1,,A∗∼B∗

  2. ∣A∣=∣B∣,∑i=1nAii=∑i=1nbii,r(A)=r(B)|A| = |B|,\sum_{i = 1}^{n}A_{{ii}} = \sum_{i =1}^{n}b_{{ii}},r(A) = r(B)∣A∣=∣B∣,∑i=1n​Aii​=∑i=1n​bii​,r(A)=r(B)

  3. ∣λE−A∣=∣λE−B∣|\lambda E - A| = |\lambda E - B|∣λE−A∣=∣λE−B∣,对∀λ\forall\lambda∀λ成立

3.矩阵可相似对角化的充分必要条件

(1)设AAA为nnn阶方阵,则AAA可对角化⇔\Leftrightarrow⇔对每个kik_{i}ki​重根特征值λi\lambda_{i}λi​,有n−r(λiE−A)=kin-r(\lambda_{i}E - A) = k_{i}n−r(λi​E−A)=ki​

(2) 设AAA可对角化,则由P−1AP=Λ,P^{- 1}{AP} = \Lambda,P−1AP=Λ,有A=PΛP−1A = {PΛ}P^{-1}A=PΛP−1,从而An=PΛnP−1A^{n} = P\Lambda^{n}P^{- 1}An=PΛnP−1

(3) 重要结论

  1. 若A∼B,C∼DA \sim B,C \sim DA∼B,C∼D,则[AOOC]∼[BOOD]\begin{bmatrix} A & O \\ O & C \\\end{bmatrix} \sim \begin{bmatrix} B & O \\ O & D \\\end{bmatrix}[AO​OC​]∼[BO​OD​].

  2. 若A∼BA \sim BA∼B,则f(A)∼f(B),∣f(A)∣∼∣f(B)∣f(A) \sim f(B),\left| f(A) \right| \sim \left| f(B)\right|f(A)∼f(B),∣f(A)∣∼∣f(B)∣,其中f(A)f(A)f(A)为关于nnn阶方阵AAA的多项式。

  3. 若AAA为可对角化矩阵,则其非零特征值的个数(重根重复计算)=秩(AAA)

4.实对称矩阵的特征值、特征向量及相似对角阵

(1)相似矩阵:设A,BA,BA,B为两个nnn阶方阵,如果存在一个可逆矩阵PPP,使得B=P−1APB =P^{- 1}{AP}B=P−1AP成立,则称矩阵AAA与BBB相似,记为A∼BA \sim BA∼B。

(2)相似矩阵的性质:如果A∼BA \sim BA∼B则有:

  1. AT∼BTA^{T} \sim B^{T}AT∼BT

  2. A−1∼B−1A^{- 1} \sim B^{- 1}A−1∼B−1 (若AAA,BBB均可逆)

  3. Ak∼BkA^{k} \sim B^{k}Ak∼Bk (kkk为正整数)

  4. ∣λE−A∣=∣λE−B∣\left| {λE} - A \right| = \left| {λE} - B \right|∣λE−A∣=∣λE−B∣,从而A,BA,BA,B 有相同的特征值

  5. ∣A∣=∣B∣\left| A \right| = \left| B \right|∣A∣=∣B∣,从而A,BA,BA,B同时可逆或者不可逆

  6. 秩(A)=\left( A \right) =(A)=秩(B),∣λE−A∣=∣λE−B∣\left( B \right),\left| {λE} - A \right| =\left| {λE} - B \right|(B),∣λE−A∣=∣λE−B∣,A,BA,BA,B不一定相似

二次型

1.n\mathbf{n}n个变量x1,x2,⋯ ,xn\mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}\mathbf{,\cdots,}\mathbf{x}_{\mathbf{n}}x1​,x2​,⋯,xn​的二次齐次函数

f(x1,x2,⋯ ,xn)=∑i=1n∑j=1naijxiyjf(x_{1},x_{2},\cdots,x_{n}) = \sum_{i = 1}^{n}{\sum_{j =1}^{n}{a_{{ij}}x_{i}y_{j}}}f(x1​,x2​,⋯,xn​)=∑i=1n​∑j=1n​aij​xi​yj​,其中aij=aji(i,j=1,2,⋯ ,n)a_{{ij}} = a_{{ji}}(i,j =1,2,\cdots,n)aij​=aji​(i,j=1,2,⋯,n),称为nnn元二次型,简称二次型. 若令x= [x1x1⋮xn],A=[a11a12⋯a1na21a22⋯a2n⋯⋯⋯⋯an1an2⋯ann]x = \ \begin{bmatrix}x_{1} \\ x_{1} \\ \vdots \\ x_{n} \\ \end{bmatrix},A = \begin{bmatrix} a_{11}& a_{12}& \cdots & a_{1n} \\ a_{21}& a_{22}& \cdots & a_{2n} \\ \cdots &\cdots &\cdots &\cdots \\ a_{n1}& a_{n2} & \cdots & a_{{nn}} \\\end{bmatrix}x= ⎣⎢⎢⎢⎢⎡​x1​x1​⋮xn​​⎦⎥⎥⎥⎥⎤​,A=⎣⎢⎢⎢⎡​a11​a21​⋯an1​​a12​a22​⋯an2​​⋯⋯⋯⋯​a1n​a2n​⋯ann​​⎦⎥⎥⎥⎤​,这二次型fff可改写成矩阵向量形式f=xTAxf =x^{T}{Ax}f=xTAx。其中AAA称为二次型矩阵,因为aij=aji(i,j=1,2,⋯ ,n)a_{{ij}} =a_{{ji}}(i,j =1,2,\cdots,n)aij​=aji​(i,j=1,2,⋯,n),所以二次型矩阵均为对称矩阵,且二次型与对称矩阵一一对应,并把矩阵AAA的秩称为二次型的秩。

2.惯性定理,二次型的标准形和规范形

(1) 惯性定理

对于任一二次型,不论选取怎样的合同变换使它化为仅含平方项的标准型,其正负惯性指数与所选变换无关,这就是所谓的惯性定理。

(2) 标准形

二次型f=(x1,x2,⋯ ,xn)=xTAxf = \left( x_{1},x_{2},\cdots,x_{n} \right) =x^{T}{Ax}f=(x1​,x2​,⋯,xn​)=xTAx经过合同变换x=Cyx = {Cy}x=Cy化为f=xTAx=yTCTACf = x^{T}{Ax} =y^{T}C^{T}{AC}f=xTAx=yTCTAC

y=∑i=1rdiyi2y = \sum_{i = 1}^{r}{d_{i}y_{i}^{2}}y=∑i=1r​di​yi2​称为 f(r≤n)f(r \leq n)f(r≤n)的标准形。在一般的数域内,二次型的标准形不是唯一的,与所作的合同变换有关,但系数不为零的平方项的个数由r(A)r(A)r(A)唯一确定。

(3) 规范形

任一实二次型fff都可经过合同变换化为规范形f=z12+z22+⋯zp2−zp+12−⋯−zr2f = z_{1}^{2} + z_{2}^{2} + \cdots z_{p}^{2} - z_{p + 1}^{2} - \cdots -z_{r}^{2}f=z12​+z22​+⋯zp2​−zp+12​−⋯−zr2​,其中rrr为AAA的秩,ppp为正惯性指数,r−pr -pr−p为负惯性指数,且规范型唯一。

3.用正交变换和配方法化二次型为标准形,二次型及其矩阵的正定性

设AAA正定⇒kA(k>0),AT,A−1,A∗\Rightarrow {kA}(k > 0),A^{T},A^{- 1},A^{*}⇒kA(k>0),AT,A−1,A∗正定;∣A∣>0|A| >0∣A∣>0,AAA可逆;aii>0a_{{ii}} > 0aii​>0,且∣Aii∣>0|A_{{ii}}| > 0∣Aii​∣>0

AAA,BBB正定⇒A+B\Rightarrow A +B⇒A+B正定,但AB{AB}AB,BA{BA}BA不一定正定

AAA正定⇔f(x)=xTAx>0,∀x≠0\Leftrightarrow f(x) = x^{T}{Ax} > 0,\forall x \neq 0⇔f(x)=xTAx>0,∀x=0

⇔A\Leftrightarrow A⇔A的各阶顺序主子式全大于零

⇔A\Leftrightarrow A⇔A的所有特征值大于零

⇔A\Leftrightarrow A⇔A的正惯性指数为nnn

⇔\Leftrightarrow⇔存在可逆阵PPP使A=PTPA = P^{T}PA=PTP

⇔\Leftrightarrow⇔存在正交矩阵QQQ,使QTAQ=Q−1AQ=(λ1⋱λn),Q^{T}{AQ} = Q^{- 1}{AQ} =\begin{pmatrix} \lambda_{1} & & \\ \begin{matrix} & \\ & \\ \end{matrix} &\ddots & \\ & & \lambda_{n} \\ \end{pmatrix},QTAQ=Q−1AQ=⎝⎜⎜⎜⎛​λ1​​​​⋱​λn​​⎠⎟⎟⎟⎞​,

其中λi>0,i=1,2,⋯ ,n.\lambda_{i} > 0,i = 1,2,\cdots,n.λi​>0,i=1,2,⋯,n.正定⇒kA(k>0),AT,A−1,A∗\Rightarrow {kA}(k >0),A^{T},A^{- 1},A^{*}⇒kA(k>0),AT,A−1,A∗正定; ∣A∣>0,A|A| > 0,A∣A∣>0,A可逆;aii>0a_{{ii}} >0aii​>0,且∣Aii∣>0|A_{{ii}}| > 0∣Aii​∣>0 。

概率论和数理统计

随机事件和概率

1.事件的关系与运算

(1) 子事件:A⊂BA \subset BA⊂B,若AAA发生,则BBB发生。

(2) 相等事件:A=BA = BA=B,即A⊂BA \subset BA⊂B,且B⊂AB \subset AB⊂A 。

(3) 和事件:A⋃BA\bigcup BA⋃B(或A+BA + BA+B),AAA与BBB中至少有一个发生。

(4) 差事件:A−BA - BA−B,AAA发生但BBB不发生。

(5) 积事件:A⋂BA\bigcap BA⋂B(或AB{AB}AB),AAA与BBB同时发生。

(6) 互斥事件(互不相容):A⋂BA\bigcap BA⋂B=∅\varnothing∅。

(7) 互逆事件(对立事件): A⋂B=∅,A⋃B=Ω,A=Bˉ,B=AˉA\bigcap B=\varnothing ,A\bigcup B=\Omega ,A=\bar{B},B=\bar{A}A⋂B=∅,A⋃B=Ω,A=Bˉ,B=Aˉ 2.运算律 (1) 交换律:A⋃B=B⋃A,A⋂B=B⋂AA\bigcup B=B\bigcup A,A\bigcap B=B\bigcap AA⋃B=B⋃A,A⋂B=B⋂A (2) 结合律:(A⋃B)⋃C=A⋃(B⋃C)(A\bigcup B)\bigcup C=A\bigcup (B\bigcup C)(A⋃B)⋃C=A⋃(B⋃C) (3) 分配律:(A⋂B)⋂C=A⋂(B⋂C)(A\bigcap B)\bigcap C=A\bigcap (B\bigcap C)(A⋂B)⋂C=A⋂(B⋂C) 3.德$\centerdot $摩根律

A⋃B‾=Aˉ⋂Bˉ\overline{A\bigcup B}=\bar{A}\bigcap \bar{B}A⋃B​=Aˉ⋂Bˉ A⋂B‾=Aˉ⋃Bˉ\overline{A\bigcap B}=\bar{A}\bigcup \bar{B}A⋂B​=Aˉ⋃Bˉ 4.完全事件组

A1A2⋯An{{A}_{1}}{{A}_{2}}\cdots {{A}_{n}}A1​A2​⋯An​两两互斥,且和事件为必然事件,即${{A}{i}}\bigcap {{A}{j}}=\varnothing, i\ne j ,\underset{i=1}{\overset{n}{\mathop \bigcup }},=\Omega $

5.概率的基本公式 (1)条件概率: P(B∣A)=P(AB)P(A)P(B|A)=\frac{P(AB)}{P(A)}P(B∣A)=P(A)P(AB)​,表示AAA发生的条件下,BBB发生的概率。 (2)全概率公式: $P(A)=\sum\limits_{i=1}^{n}{P(A|{{B}{i}})P({{B}{i}}),{{B}{i}}{{B}{j}}}=\varnothing ,i\ne j,\underset{i=1}{\overset{n}{\mathop{\bigcup }}},{{B}_{i}}=\Omega $ (3) Bayes公式:

P(Bj∣A)=P(A∣Bj)P(Bj)∑i=1nP(A∣Bi)P(Bi),j=1,2,⋯ ,nP({{B}_{j}}|A)=\frac{P(A|{{B}_{j}})P({{B}_{j}})}{\sum\limits_{i=1}^{n}{P(A|{{B}_{i}})P({{B}_{i}})}},j=1,2,\cdots ,nP(Bj​∣A)=i=1∑n​P(A∣Bi​)P(Bi​)P(A∣Bj​)P(Bj​)​,j=1,2,⋯,n 注:上述公式中事件Bi{{B}_{i}}Bi​的个数可为可列个。 (4)乘法公式: P(A1A2)=P(A1)P(A2∣A1)=P(A2)P(A1∣A2)P({{A}_{1}}{{A}_{2}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})=P({{A}_{2}})P({{A}_{1}}|{{A}_{2}})P(A1​A2​)=P(A1​)P(A2​∣A1​)=P(A2​)P(A1​∣A2​) P(A1A2⋯An)=P(A1)P(A2∣A1)P(A3∣A1A2)⋯P(An∣A1A2⋯An−1)P({{A}_{1}}{{A}_{2}}\cdots {{A}_{n}})=P({{A}_{1}})P({{A}_{2}}|{{A}_{1}})P({{A}_{3}}|{{A}_{1}}{{A}_{2}})\cdots P({{A}_{n}}|{{A}_{1}}{{A}_{2}}\cdots {{A}_{n-1}})P(A1​A2​⋯An​)=P(A1​)P(A2​∣A1​)P(A3​∣A1​A2​)⋯P(An​∣A1​A2​⋯An−1​)

6.事件的独立性 (1)AAA与BBB相互独立⇔P(AB)=P(A)P(B)\Leftrightarrow P(AB)=P(A)P(B)⇔P(AB)=P(A)P(B) (2)AAA,BBB,CCC两两独立 ⇔P(AB)=P(A)P(B)\Leftrightarrow P(AB)=P(A)P(B)⇔P(AB)=P(A)P(B);P(BC)=P(B)P(C)P(BC)=P(B)P(C)P(BC)=P(B)P(C) ;P(AC)=P(A)P(C)P(AC)=P(A)P(C)P(AC)=P(A)P(C); (3)AAA,BBB,CCC相互独立 ⇔P(AB)=P(A)P(B)\Leftrightarrow P(AB)=P(A)P(B)⇔P(AB)=P(A)P(B); P(BC)=P(B)P(C)P(BC)=P(B)P(C)P(BC)=P(B)P(C) ; P(AC)=P(A)P(C)P(AC)=P(A)P(C)P(AC)=P(A)P(C) ; P(ABC)=P(A)P(B)P(C)P(ABC)=P(A)P(B)P(C)P(ABC)=P(A)P(B)P(C)

7.独立重复试验

将某试验独立重复nnn次,若每次实验中事件A发生的概率为ppp,则nnn次试验中AAA发生kkk次的概率为: P(X=k)=Cnkpk(1−p)n−kP(X=k)=C_{n}^{k}{{p}^{k}}{{(1-p)}^{n-k}}P(X=k)=Cnk​pk(1−p)n−k

8.重要公式与结论

(1)P(Aˉ)=1−P(A)(1)P(\bar{A})=1-P(A)(1)P(Aˉ)=1−P(A)

(2)P(A⋃B)=P(A)+P(B)−P(AB)(2)P(A\bigcup B)=P(A)+P(B)-P(AB)(2)P(A⋃B)=P(A)+P(B)−P(AB) P(A⋃B⋃C)=P(A)+P(B)+P(C)−P(AB)−P(BC)−P(AC)+P(ABC)P(A\bigcup B\bigcup C)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC)P(A⋃B⋃C)=P(A)+P(B)+P(C)−P(AB)−P(BC)−P(AC)+P(ABC)

(3)P(A−B)=P(A)−P(AB)(3)P(A-B)=P(A)-P(AB)(3)P(A−B)=P(A)−P(AB)

(4)P(ABˉ)=P(A)−P(AB),P(A)=P(AB)+P(ABˉ),(4)P(A\bar{B})=P(A)-P(AB),P(A)=P(AB)+P(A\bar{B}),(4)P(ABˉ)=P(A)−P(AB),P(A)=P(AB)+P(ABˉ),

P(A⋃B)=P(A)+P(AˉB)=P(AB)+P(ABˉ)+P(AˉB)P(A\bigcup B)=P(A)+P(\bar{A}B)=P(AB)+P(A\bar{B})+P(\bar{A}B)P(A⋃B)=P(A)+P(AˉB)=P(AB)+P(ABˉ)+P(AˉB)

(5)条件概率P(⋅∣B)P(\centerdot |B)P(⋅∣B)满足概率的所有性质, 例如:. P(Aˉ1∣B)=1−P(A1∣B)P({{\bar{A}}_{1}}|B)=1-P({{A}_{1}}|B)P(Aˉ1​∣B)=1−P(A1​∣B) P(A1⋃A2∣B)=P(A1∣B)+P(A2∣B)−P(A1A2∣B)P({{A}_{1}}\bigcup {{A}_{2}}|B)=P({{A}_{1}}|B)+P({{A}_{2}}|B)-P({{A}_{1}}{{A}_{2}}|B)P(A1​⋃A2​∣B)=P(A1​∣B)+P(A2​∣B)−P(A1​A2​∣B) P(A1A2∣B)=P(A1∣B)P(A2∣A1B)P({{A}_{1}}{{A}_{2}}|B)=P({{A}_{1}}|B)P({{A}_{2}}|{{A}_{1}}B)P(A1​A2​∣B)=P(A1​∣B)P(A2​∣A1​B)

(6)若A1,A2,⋯ ,An{{A}_{1}},{{A}_{2}},\cdots ,{{A}_{n}}A1​,A2​,⋯,An​相互独立,则P(⋂i=1nAi)=∏i=1nP(Ai),P(\bigcap\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{P({{A}_{i}})},P(i=1⋂n​Ai​)=i=1∏n​P(Ai​), P(⋃i=1nAi)=∏i=1n(1−P(Ai))P(\bigcup\limits_{i=1}^{n}{{{A}_{i}}})=\prod\limits_{i=1}^{n}{(1-P({{A}_{i}}))}P(i=1⋃n​Ai​)=i=1∏n​(1−P(Ai​))

(7)互斥、互逆与独立性之间的关系: AAA与BBB互逆⇒\Rightarrow⇒ AAA与BBB互斥,但反之不成立,AAA与BBB互斥(或互逆)且均非零概率事件$\Rightarrow $$A与与与B$不独立.

(8)若A1,A2,⋯ ,Am,B1,B2,⋯ ,Bn{{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}},{{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}}A1​,A2​,⋯,Am​,B1​,B2​,⋯,Bn​相互独立,则f(A1,A2,⋯ ,Am)f({{A}_{1}},{{A}_{2}},\cdots ,{{A}_{m}})f(A1​,A2​,⋯,Am​)与g(B1,B2,⋯ ,Bn)g({{B}_{1}},{{B}_{2}},\cdots ,{{B}_{n}})g(B1​,B2​,⋯,Bn​)也相互独立,其中f(⋅),g(⋅)f(\centerdot ),g(\centerdot )f(⋅),g(⋅)分别表示对相应事件做任意事件运算后所得的事件,另外,概率为1(或0)的事件与任何事件相互独立.

随机变量及其概率分布

1.随机变量及概率分布

取值带有随机性的变量,严格地说是定义在样本空间上,取值于实数的函数称为随机变量,概率分布通常指分布函数或分布律

2.分布函数的概念与性质

定义: F(x)=P(X≤x),−∞

性质:(1)0≤F(x)≤10 \leq F(x) \leq 10≤F(x)≤1

(2) F(x)F(x)F(x)单调不减

(3) 右连续F(x+0)=F(x)F(x + 0) = F(x)F(x+0)=F(x)

(4) F(−∞)=0,F(+∞)=1F( - \infty) = 0,F( + \infty) = 1F(−∞)=0,F(+∞)=1

3.离散型随机变量的概率分布

P(X=xi)=pi,i=1,2,⋯ ,n,⋯pi≥0,∑i=1∞pi=1P(X = x_{i}) = p_{i},i = 1,2,\cdots,n,\cdots\quad\quad p_{i} \geq 0,\sum_{i =1}^{\infty}p_{i} = 1P(X=xi​)=pi​,i=1,2,⋯,n,⋯pi​≥0,∑i=1∞​pi​=1

4.连续型随机变量的概率密度

概率密度f(x)f(x)f(x);非负可积,且:

(1)f(x)≥0,f(x) \geq 0,f(x)≥0,

(2)∫−∞+∞f(x)dx=1\int_{- \infty}^{+\infty}{f(x){dx} = 1}∫−∞+∞​f(x)dx=1

(3)xxx为f(x)f(x)f(x)的连续点,则:

f(x)=F′(x)f(x) = F'(x)f(x)=F′(x)分布函数F(x)=∫−∞xf(t)dtF(x) = \int_{- \infty}^{x}{f(t){dt}}F(x)=∫−∞x​f(t)dt

5.常见分布

(1) 0-1分布:P(X=k)=pk(1−p)1−k,k=0,1P(X = k) = p^{k}{(1 - p)}^{1 - k},k = 0,1P(X=k)=pk(1−p)1−k,k=0,1

(2) 二项分布:B(n,p)B(n,p)B(n,p): P(X=k)=Cnkpk(1−p)n−k,k=0,1,⋯ ,nP(X = k) = C_{n}^{k}p^{k}{(1 - p)}^{n - k},k =0,1,\cdots,nP(X=k)=Cnk​pk(1−p)n−k,k=0,1,⋯,n

(3) Poisson分布:p(λ)p(\lambda)p(λ): P(X=k)=λkk!e−λ,λ>0,k=0,1,2⋯P(X = k) = \frac{\lambda^{k}}{k!}e^{-\lambda},\lambda > 0,k = 0,1,2\cdotsP(X=k)=k!λk​e−λ,λ>0,k=0,1,2⋯

(4) 均匀分布U(a,b)U(a,b)U(a,b):$f(x) = { \begin{matrix} & \frac{1}{b - a},a < x< b \ & 0, \ \end{matrix} $

(5) 正态分布:N(μ,σ2):N(\mu,\sigma^{2}):N(μ,σ2): φ(x)=12πσe−(x−μ)22σ2,σ>0,∞ 0,\infty < x < + \inftyφ(x)=2π​σ1​e−2σ2(x−μ)2​,σ>0,∞

(6)指数分布:$E(\lambda):f(x) ={ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \ & 0, \ \end{matrix} $

(7)几何分布:G(p):P(X=k)=(1−p)k−1p,0

(8)超几何分布: H(N,M,n):P(X=k)=CMkCN−Mn−kCNn,k=0,1,⋯ ,min(n,M)H(N,M,n):P(X = k) = \frac{C_{M}^{k}C_{N - M}^{n -k}}{C_{N}^{n}},k =0,1,\cdots,min(n,M)H(N,M,n):P(X=k)=CNn​CMk​CN−Mn−k​​,k=0,1,⋯,min(n,M)

6.随机变量函数的概率分布

(1)离散型:P(X=x1)=pi,Y=g(X)P(X = x_{1}) = p_{i},Y = g(X)P(X=x1​)=pi​,Y=g(X)

则: P(Y=yj)=∑g(xi)=yiP(X=xi)P(Y = y_{j}) = \sum_{g(x_{i}) = y_{i}}^{}{P(X = x_{i})}P(Y=yj​)=∑g(xi​)=yi​​P(X=xi​)

(2)连续型:X ~fX(x),Y=g(x)X\tilde{\ }f_{X}(x),Y = g(x)X ~fX​(x),Y=g(x)

则:Fy(y)=P(Y≤y)=P(g(X)≤y)=∫g(x)≤yfx(x)dxF_{y}(y) = P(Y \leq y) = P(g(X) \leq y) = \int_{g(x) \leq y}^{}{f_{x}(x)dx}Fy​(y)=P(Y≤y)=P(g(X)≤y)=∫g(x)≤y​fx​(x)dx, fY(y)=FY′(y)f_{Y}(y) = F'_{Y}(y)fY​(y)=FY′​(y)

7.重要公式与结论

(1) X∼N(0,1)⇒φ(0)=12π,Φ(0)=12,X\sim N(0,1) \Rightarrow \varphi(0) = \frac{1}{\sqrt{2\pi}},\Phi(0) =\frac{1}{2},X∼N(0,1)⇒φ(0)=2π​1​,Φ(0)=21​, Φ(−a)=P(X≤−a)=1−Φ(a)\Phi( - a) = P(X \leq - a) = 1 - \Phi(a)Φ(−a)=P(X≤−a)=1−Φ(a)

(2) X∼N(μ,σ2)⇒X−μσ∼N(0,1),P(X≤a)=Φ(a−μσ)X\sim N\left( \mu,\sigma^{2} \right) \Rightarrow \frac{X -\mu}{\sigma}\sim N\left( 0,1 \right),P(X \leq a) = \Phi(\frac{a -\mu}{\sigma})X∼N(μ,σ2)⇒σX−μ​∼N(0,1),P(X≤a)=Φ(σa−μ​)

(3) X∼E(λ)⇒P(X>s+t∣X>s)=P(X>t)X\sim E(\lambda) \Rightarrow P(X > s + t|X > s) = P(X > t)X∼E(λ)⇒P(X>s+t∣X>s)=P(X>t)

(4) X∼G(p)⇒P(X=m+k∣X>m)=P(X=k)X\sim G(p) \Rightarrow P(X = m + k|X > m) = P(X = k)X∼G(p)⇒P(X=m+k∣X>m)=P(X=k)

(5) 离散型随机变量的分布函数为阶梯间断函数;连续型随机变量的分布函数为连续函数,但不一定为处处可导函数。

(6) 存在既非离散也非连续型随机变量。

多维随机变量及其分布

1.二维随机变量及其联合分布

由两个随机变量构成的随机向量(X,Y)(X,Y)(X,Y), 联合分布为F(x,y)=P(X≤x,Y≤y)F(x,y) = P(X \leq x,Y \leq y)F(x,y)=P(X≤x,Y≤y)

2.二维离散型随机变量的分布

(1) 联合概率分布律 P{X=xi,Y=yj}=pij;i,j=1,2,⋯P\{ X = x_{i},Y = y_{j}\} = p_{{ij}};i,j =1,2,\cdotsP{X=xi​,Y=yj​}=pij​;i,j=1,2,⋯

(2) 边缘分布律 pi⋅=∑j=1∞pij,i=1,2,⋯p_{i \cdot} = \sum_{j = 1}^{\infty}p_{{ij}},i =1,2,\cdotspi⋅​=∑j=1∞​pij​,i=1,2,⋯ p⋅j=∑i∞pij,j=1,2,⋯p_{\cdot j} = \sum_{i}^{\infty}p_{{ij}},j = 1,2,\cdotsp⋅j​=∑i∞​pij​,j=1,2,⋯

(3) 条件分布律 P{X=xi∣Y=yj}=pijp⋅jP\{ X = x_{i}|Y = y_{j}\} = \frac{p_{{ij}}}{p_{\cdot j}}P{X=xi​∣Y=yj​}=p⋅j​pij​​ P{Y=yj∣X=xi}=pijpi⋅P\{ Y = y_{j}|X = x_{i}\} = \frac{p_{{ij}}}{p_{i \cdot}}P{Y=yj​∣X=xi​}=pi⋅​pij​​

3. 二维连续性随机变量的密度

(1) 联合概率密度f(x,y):f(x,y):f(x,y):

  1. f(x,y)≥0f(x,y) \geq 0f(x,y)≥0

  2. ∫−∞+∞∫−∞+∞f(x,y)dxdy=1\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{f(x,y)dxdy}} = 1∫−∞+∞​∫−∞+∞​f(x,y)dxdy=1

(2) 分布函数:F(x,y)=∫−∞x∫−∞yf(u,v)dudvF(x,y) = \int_{- \infty}^{x}{\int_{- \infty}^{y}{f(u,v)dudv}}F(x,y)=∫−∞x​∫−∞y​f(u,v)dudv

(3) 边缘概率密度: fX(x)=∫−∞+∞f(x,y)dyf_{X}\left( x \right) = \int_{- \infty}^{+ \infty}{f\left( x,y \right){dy}}fX​(x)=∫−∞+∞​f(x,y)dy fY(y)=∫−∞+∞f(x,y)dxf_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}fY​(y)=∫−∞+∞​f(x,y)dx

(4) 条件概率密度:fX∣Y(x|y)=f(x,y)fY(y)f_{X|Y}\left( x \middle| y \right) = \frac{f\left( x,y \right)}{f_{Y}\left( y \right)}fX∣Y​(x∣y)=fY​(y)f(x,y)​ fY∣X(y∣x)=f(x,y)fX(x)f_{Y|X}(y|x) = \frac{f(x,y)}{f_{X}(x)}fY∣X​(y∣x)=fX​(x)f(x,y)​

4.常见二维随机变量的联合分布

(1) 二维均匀分布:(x,y)∼U(D)(x,y) \sim U(D)(x,y)∼U(D) ,f(x,y)={1S(D),(x,y)∈D0,其他f(x,y) = \begin{cases} \frac{1}{S(D)},(x,y) \in D \\ 0,其他 \end{cases}f(x,y)={S(D)1​,(x,y)∈D0,其他​

(2) 二维正态分布:(X,Y)∼N(μ1,μ2,σ12,σ22,ρ)(X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)(X,Y)∼N(μ1​,μ2​,σ12​,σ22​,ρ),(X,Y)∼N(μ1,μ2,σ12,σ22,ρ)(X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)(X,Y)∼N(μ1​,μ2​,σ12​,σ22​,ρ)

f(x,y)=12πσ1σ21−ρ2.exp⁡{−12(1−ρ2)[(x−μ1)2σ12−2ρ(x−μ1)(y−μ2)σ1σ2+(y−μ2)2σ22]}f(x,y) = \frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1 - \rho^{2}}}.\exp\left\{ \frac{- 1}{2(1 - \rho^{2})}\lbrack\frac{{(x - \mu_{1})}^{2}}{\sigma_{1}^{2}} - 2\rho\frac{(x - \mu_{1})(y - \mu_{2})}{\sigma_{1}\sigma_{2}} + \frac{{(y - \mu_{2})}^{2}}{\sigma_{2}^{2}}\rbrack \right\}f(x,y)=2πσ1​σ2​1−ρ2​1​.exp{2(1−ρ2)−1​[σ12​(x−μ1​)2​−2ρσ1​σ2​(x−μ1​)(y−μ2​)​+σ22​(y−μ2​)2​]}

5.随机变量的独立性和相关性

XXX和YYY的相互独立:⇔F(x,y)=FX(x)FY(y)\Leftrightarrow F\left( x,y \right) = F_{X}\left( x \right)F_{Y}\left( y \right)⇔F(x,y)=FX​(x)FY​(y):

⇔pij=pi⋅⋅p⋅j\Leftrightarrow p_{{ij}} = p_{i \cdot} \cdot p_{\cdot j}⇔pij​=pi⋅​⋅p⋅j​(离散型) ⇔f(x,y)=fX(x)fY(y)\Leftrightarrow f\left( x,y \right) = f_{X}\left( x \right)f_{Y}\left( y \right)⇔f(x,y)=fX​(x)fY​(y)(连续型)

XXX和YYY的相关性:

相关系数ρXY=0\rho_{{XY}} = 0ρXY​=0时,称XXX和YYY不相关, 否则称XXX和YYY相关

6.两个随机变量简单函数的概率分布

离散型: P(X=xi,Y=yi)=pij,Z=g(X,Y)P\left( X = x_{i},Y = y_{i} \right) = p_{{ij}},Z = g\left( X,Y \right)P(X=xi​,Y=yi​)=pij​,Z=g(X,Y) 则:

P(Z=zk)=P{g(X,Y)=zk}=∑g(xi,yi)=zkP(X=xi,Y=yj)P(Z = z_{k}) = P\left\{ g\left( X,Y \right) = z_{k} \right\} = \sum_{g\left( x_{i},y_{i} \right) = z_{k}}^{}{P\left( X = x_{i},Y = y_{j} \right)}P(Z=zk​)=P{g(X,Y)=zk​}=∑g(xi​,yi​)=zk​​P(X=xi​,Y=yj​)

连续型: (X,Y)∼f(x,y),Z=g(X,Y)\left( X,Y \right) \sim f\left( x,y \right),Z = g\left( X,Y \right)(X,Y)∼f(x,y),Z=g(X,Y) 则:

Fz(z)=P{g(X,Y)≤z}=∬g(x,y)≤zf(x,y)dxdyF_{z}\left( z \right) = P\left\{ g\left( X,Y \right) \leq z \right\} = \iint_{g(x,y) \leq z}^{}{f(x,y)dxdy}Fz​(z)=P{g(X,Y)≤z}=∬g(x,y)≤z​f(x,y)dxdy,fz(z)=Fz′(z)f_{z}(z) = F'_{z}(z)fz​(z)=Fz′​(z)

7.重要公式与结论

(1) 边缘密度公式: fX(x)=∫−∞+∞f(x,y)dy,f_{X}(x) = \int_{- \infty}^{+ \infty}{f(x,y)dy,}fX​(x)=∫−∞+∞​f(x,y)dy, fY(y)=∫−∞+∞f(x,y)dxf_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}fY​(y)=∫−∞+∞​f(x,y)dx

(2) P{(X,Y)∈D}=∬Df(x,y)dxdyP\left\{ \left( X,Y \right) \in D \right\} = \iint_{D}^{}{f\left( x,y \right){dxdy}}P{(X,Y)∈D}=∬D​f(x,y)dxdy

(3) 若(X,Y)(X,Y)(X,Y)服从二维正态分布N(μ1,μ2,σ12,σ22,ρ)N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)N(μ1​,μ2​,σ12​,σ22​,ρ) 则有:

  1. X∼N(μ1,σ12),Y∼N(μ2,σ22).X\sim N\left( \mu_{1},\sigma_{1}^{2} \right),Y\sim N(\mu_{2},\sigma_{2}^{2}).X∼N(μ1​,σ12​),Y∼N(μ2​,σ22​).

  2. XXX与YYY相互独立⇔ρ=0\Leftrightarrow \rho = 0⇔ρ=0,即XXX与YYY不相关。

  3. C1X+C2Y∼N(C1μ1+C2μ2,C12σ12+C22σ22+2C1C2σ1σ2ρ)C_{1}X + C_{2}Y\sim N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} + C_{2}^{2}\sigma_{2}^{2} + 2C_{1}C_{2}\sigma_{1}\sigma_{2}\rho)C1​X+C2​Y∼N(C1​μ1​+C2​μ2​,C12​σ12​+C22​σ22​+2C1​C2​σ1​σ2​ρ)

  4.  X{\ X} X关于Y=yY=yY=y的条件分布为: N(μ1+ρσ1σ2(y−μ2),σ12(1−ρ2))N(\mu_{1} + \rho\frac{\sigma_{1}}{\sigma_{2}}(y - \mu_{2}),\sigma_{1}^{2}(1 - \rho^{2}))N(μ1​+ρσ2​σ1​​(y−μ2​),σ12​(1−ρ2))

  5. YYY关于X=xX = xX=x的条件分布为: N(μ2+ρσ2σ1(x−μ1),σ22(1−ρ2))N(\mu_{2} + \rho\frac{\sigma_{2}}{\sigma_{1}}(x - \mu_{1}),\sigma_{2}^{2}(1 - \rho^{2}))N(μ2​+ρσ1​σ2​​(x−μ1​),σ22​(1−ρ2))

(4) 若XXX与YYY独立,且分别服从N(μ1,σ12),N(μ1,σ22),N(\mu_{1},\sigma_{1}^{2}),N(\mu_{1},\sigma_{2}^{2}),N(μ1​,σ12​),N(μ1​,σ22​), 则:(X,Y)∼N(μ1,μ2,σ12,σ22,0),\left( X,Y \right)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},0),(X,Y)∼N(μ1​,μ2​,σ12​,σ22​,0),

C1X+C2Y ~N(C1μ1+C2μ2,C12σ12C22σ22).C_{1}X + C_{2}Y\tilde{\ }N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} C_{2}^{2}\sigma_{2}^{2}).C1​X+C2​Y ~N(C1​μ1​+C2​μ2​,C12​σ12​C22​σ22​).

(5) 若XXX与YYY相互独立,f(x)f\left( x \right)f(x)和g(x)g\left( x \right)g(x)为连续函数, 则f(X)f\left( X \right)f(X)和g(Y)g(Y)g(Y)也相互独立。

随机变量的数字特征

1.数学期望

离散型:P{X=xi}=pi,E(X)=∑ixipiP\left\{ X = x_{i} \right\} = p_{i},E(X) = \sum_{i}^{}{x_{i}p_{i}}P{X=xi​}=pi​,E(X)=∑i​xi​pi​;

连续型: X∼f(x),E(X)=∫−∞+∞xf(x)dxX\sim f(x),E(X) = \int_{- \infty}^{+ \infty}{xf(x)dx}X∼f(x),E(X)=∫−∞+∞​xf(x)dx

性质:

(1) E(C)=C,E[E(X)]=E(X)E(C) = C,E\lbrack E(X)\rbrack = E(X)E(C)=C,E[E(X)]=E(X)

(2) E(C1X+C2Y)=C1E(X)+C2E(Y)E(C_{1}X + C_{2}Y) = C_{1}E(X) + C_{2}E(Y)E(C1​X+C2​Y)=C1​E(X)+C2​E(Y)

(3) 若XXX和YYY独立,则E(XY)=E(X)E(Y)E(XY) = E(X)E(Y)E(XY)=E(X)E(Y)

(4)[E(XY)]2≤E(X2)E(Y2)\left\lbrack E(XY) \right\rbrack^{2} \leq E(X^{2})E(Y^{2})[E(XY)]2≤E(X2)E(Y2)

2.方差:D(X)=E[X−E(X)]2=E(X2)−[E(X)]2D(X) = E\left\lbrack X - E(X) \right\rbrack^{2} = E(X^{2}) - \left\lbrack E(X) \right\rbrack^{2}D(X)=E[X−E(X)]2=E(X2)−[E(X)]2

3.标准差:D(X)\sqrt{D(X)}D(X)​,

4.离散型:D(X)=∑i[xi−E(X)]2piD(X) = \sum_{i}^{}{\left\lbrack x_{i} - E(X) \right\rbrack^{2}p_{i}}D(X)=∑i​[xi​−E(X)]2pi​

5.连续型:D(X)=∫−∞+∞[x−E(X)]2f(x)dxD(X) = {\int_{- \infty}^{+ \infty}\left\lbrack x - E(X) \right\rbrack}^{2}f(x)dxD(X)=∫−∞+∞​[x−E(X)]2f(x)dx

性质:

(1) D(C)=0,D[E(X)]=0,D[D(X)]=0\ D(C) = 0,D\lbrack E(X)\rbrack = 0,D\lbrack D(X)\rbrack = 0 D(C)=0,D[E(X)]=0,D[D(X)]=0

(2) XXX与YYY相互独立,则D(X±Y)=D(X)+D(Y)D(X \pm Y) = D(X) + D(Y)D(X±Y)=D(X)+D(Y)

(3) D(C1X+C2)=C12D(X)\ D\left( C_{1}X + C_{2} \right) = C_{1}^{2}D\left( X \right) D(C1​X+C2​)=C12​D(X)

(4) 一般有 D(X±Y)=D(X)+D(Y)±2Cov(X,Y)=D(X)+D(Y)±2ρD(X)D(Y)D(X \pm Y) = D(X) + D(Y) \pm 2Cov(X,Y) = D(X) + D(Y) \pm 2\rho\sqrt{D(X)}\sqrt{D(Y)}D(X±Y)=D(X)+D(Y)±2Cov(X,Y)=D(X)+D(Y)±2ρD(X)​D(Y)​

(5) D(X)

(6) D(X)=0⇔P{X=C}=1\ D(X) = 0 \Leftrightarrow P\left\{ X = C \right\} = 1 D(X)=0⇔P{X=C}=1

6.随机变量函数的数学期望

(1) 对于函数Y=g(x)Y = g(x)Y=g(x)

XXX为离散型:P{X=xi}=pi,E(Y)=∑ig(xi)piP\{ X = x_{i}\} = p_{i},E(Y) = \sum_{i}^{}{g(x_{i})p_{i}}P{X=xi​}=pi​,E(Y)=∑i​g(xi​)pi​;

XXX为连续型:X∼f(x),E(Y)=∫−∞+∞g(x)f(x)dxX\sim f(x),E(Y) = \int_{- \infty}^{+ \infty}{g(x)f(x)dx}X∼f(x),E(Y)=∫−∞+∞​g(x)f(x)dx

(2) Z=g(X,Y)Z = g(X,Y)Z=g(X,Y);(X,Y)∼P{X=xi,Y=yj}=pij\left( X,Y \right)\sim P\{ X = x_{i},Y = y_{j}\} = p_{{ij}}(X,Y)∼P{X=xi​,Y=yj​}=pij​; E(Z)=∑i∑jg(xi,yj)pijE(Z) = \sum_{i}^{}{\sum_{j}^{}{g(x_{i},y_{j})p_{{ij}}}}E(Z)=∑i​∑j​g(xi​,yj​)pij​ (X,Y)∼f(x,y)\left( X,Y \right)\sim f(x,y)(X,Y)∼f(x,y);E(Z)=∫−∞+∞∫−∞+∞g(x,y)f(x,y)dxdyE(Z) = \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{g(x,y)f(x,y)dxdy}}E(Z)=∫−∞+∞​∫−∞+∞​g(x,y)f(x,y)dxdy

7.协方差

Cov(X,Y)=E[(X−E(X)(Y−E(Y))]Cov(X,Y) = E\left\lbrack (X - E(X)(Y - E(Y)) \right\rbrackCov(X,Y)=E[(X−E(X)(Y−E(Y))]

8.相关系数

ρXY=Cov(X,Y)D(X)D(Y)\rho_{{XY}} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}ρXY​=D(X)​D(Y)​Cov(X,Y)​,kkk阶原点矩 E(Xk)E(X^{k})E(Xk); kkk阶中心矩 E{[X−E(X)]k}E\left\{ {\lbrack X - E(X)\rbrack}^{k} \right\}E{[X−E(X)]k}

性质:

(1) Cov(X,Y)=Cov(Y,X)\ Cov(X,Y) = Cov(Y,X) Cov(X,Y)=Cov(Y,X)

(2) Cov(aX,bY)=abCov(Y,X)\ Cov(aX,bY) = abCov(Y,X) Cov(aX,bY)=abCov(Y,X)

(3) Cov(X1+X2,Y)=Cov(X1,Y)+Cov(X2,Y)\ Cov(X_{1} + X_{2},Y) = Cov(X_{1},Y) + Cov(X_{2},Y) Cov(X1​+X2​,Y)=Cov(X1​,Y)+Cov(X2​,Y)

(4) ∣ρ(X,Y)∣≤1\ \left| \rho\left( X,Y \right) \right| \leq 1 ∣ρ(X,Y)∣≤1

(5)  ρ(X,Y)=1⇔P(Y=aX+b)=1\ \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1⇔P(Y=aX+b)=1 ,其中a>0a > 0a>0

ρ(X,Y)=−1⇔P(Y=aX+b)=1\rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1ρ(X,Y)=−1⇔P(Y=aX+b)=1 ,其中a<0a < 0a<0

9.重要公式与结论

(1) D(X)=E(X2)−E2(X)\ D(X) = E(X^{2}) - E^{2}(X) D(X)=E(X2)−E2(X)

(2) Cov(X,Y)=E(XY)−E(X)E(Y)\ Cov(X,Y) = E(XY) - E(X)E(Y) Cov(X,Y)=E(XY)−E(X)E(Y)

(3) ∣ρ(X,Y)∣≤1,\left| \rho\left( X,Y \right) \right| \leq 1,∣ρ(X,Y)∣≤1,且 ρ(X,Y)=1⇔P(Y=aX+b)=1\rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1ρ(X,Y)=1⇔P(Y=aX+b)=1,其中a>0a > 0a>0

ρ(X,Y)=−1⇔P(Y=aX+b)=1\rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1ρ(X,Y)=−1⇔P(Y=aX+b)=1,其中a<0a < 0a<0

(4) 下面5个条件互为充要条件:

ρ(X,Y)=0\rho(X,Y) = 0ρ(X,Y)=0 ⇔Cov(X,Y)=0\Leftrightarrow Cov(X,Y) = 0⇔Cov(X,Y)=0 ⇔E(X,Y)=E(X)E(Y)\Leftrightarrow E(X,Y) = E(X)E(Y)⇔E(X,Y)=E(X)E(Y) ⇔D(X+Y)=D(X)+D(Y)\Leftrightarrow D(X + Y) = D(X) + D(Y)⇔D(X+Y)=D(X)+D(Y) ⇔D(X−Y)=D(X)+D(Y)\Leftrightarrow D(X - Y) = D(X) + D(Y)⇔D(X−Y)=D(X)+D(Y)

注:XXX与YYY独立为上述5个条件中任何一个成立的充分条件,但非必要条件。

数理统计的基本概念

1.基本概念

总体:研究对象的全体,它是一个随机变量,用XXX表示。

个体:组成总体的每个基本元素。

简单随机样本:来自总体XXX的nnn个相互独立且与总体同分布的随机变量X1,X2⋯ ,XnX_{1},X_{2}\cdots,X_{n}X1​,X2​⋯,Xn​,称为容量为nnn的简单随机样本,简称样本。

统计量:设X1,X2⋯ ,Xn,X_{1},X_{2}\cdots,X_{n},X1​,X2​⋯,Xn​,是来自总体XXX的一个样本,g(X1,X2⋯ ,Xn)g(X_{1},X_{2}\cdots,X_{n})g(X1​,X2​⋯,Xn​))是样本的连续函数,且g()g()g()中不含任何未知参数,则称g(X1,X2⋯ ,Xn)g(X_{1},X_{2}\cdots,X_{n})g(X1​,X2​⋯,Xn​)为统计量。

样本均值:X‾=1n∑i=1nXi\overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i}X=n1​∑i=1n​Xi​

样本方差:S2=1n−1∑i=1n(Xi−X‾)2S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{2}S2=n−11​∑i=1n​(Xi​−X)2

样本矩:样本kkk阶原点矩:Ak=1n∑i=1nXik,k=1,2,⋯A_{k} = \frac{1}{n}\sum_{i = 1}^{n}X_{i}^{k},k = 1,2,\cdotsAk​=n1​∑i=1n​Xik​,k=1,2,⋯

样本kkk阶中心矩:Bk=1n∑i=1n(Xi−X‾)k,k=1,2,⋯B_{k} = \frac{1}{n}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{k},k = 1,2,\cdotsBk​=n1​∑i=1n​(Xi​−X)k,k=1,2,⋯

2.分布

χ2\chi^{2}χ2分布:χ2=X12+X22+⋯+Xn2∼χ2(n)\chi^{2} = X_{1}^{2} + X_{2}^{2} + \cdots + X_{n}^{2}\sim\chi^{2}(n)χ2=X12​+X22​+⋯+Xn2​∼χ2(n),其中X1,X2⋯ ,Xn,X_{1},X_{2}\cdots,X_{n},X1​,X2​⋯,Xn​,相互独立,且同服从N(0,1)N(0,1)N(0,1)

ttt分布:T=XY/n∼t(n)T = \frac{X}{\sqrt{Y/n}}\sim t(n)T=Y/n​X​∼t(n) ,其中X∼N(0,1),Y∼χ2(n),X\sim N\left( 0,1 \right),Y\sim\chi^{2}(n),X∼N(0,1),Y∼χ2(n),且XXX,YYY 相互独立。

FFF分布:F=X/n1Y/n2∼F(n1,n2)F = \frac{X/n_{1}}{Y/n_{2}}\sim F(n_{1},n_{2})F=Y/n2​X/n1​​∼F(n1​,n2​),其中X∼χ2(n1),Y∼χ2(n2),X\sim\chi^{2}\left( n_{1} \right),Y\sim\chi^{2}(n_{2}),X∼χ2(n1​),Y∼χ2(n2​),且XXX,YYY相互独立。

分位数:若P(X≤xα)=α,P(X \leq x_{\alpha}) = \alpha,P(X≤xα​)=α,则称xαx_{\alpha}xα​为XXX的α\alphaα分位数

3.正态总体的常用样本分布

(1) 设X1,X2⋯ ,XnX_{1},X_{2}\cdots,X_{n}X1​,X2​⋯,Xn​为来自正态总体N(μ,σ2)N(\mu,\sigma^{2})N(μ,σ2)的样本,

X‾=1n∑i=1nXi,S2=1n−1∑i=1n(Xi−X‾)2,\overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i},S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{{(X_{i} - \overline{X})}^{2},}X=n1​∑i=1n​Xi​,S2=n−11​∑i=1n​(Xi​−X)2,则:

  1. X‾∼N(μ,σ2n)  \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ }X∼N(μ,nσ2​)  或者X‾−μσn∼N(0,1)\frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)n​σ​X−μ​∼N(0,1)

  2. (n−1)S2σ2=1σ2∑i=1n(Xi−X‾)2∼χ2(n−1)\frac{(n - 1)S^{2}}{\sigma^{2}} = \frac{1}{\sigma^{2}}\sum_{i = 1}^{n}{{(X_{i} - \overline{X})}^{2}\sim\chi^{2}(n - 1)}σ2(n−1)S2​=σ21​∑i=1n​(Xi​−X)2∼χ2(n−1)

  3. 1σ2∑i=1n(Xi−μ)2∼χ2(n)\frac{1}{\sigma^{2}}\sum_{i = 1}^{n}{{(X_{i} - \mu)}^{2}\sim\chi^{2}(n)}σ21​∑i=1n​(Xi​−μ)2∼χ2(n)

4)  X‾−μS/n∼t(n−1){\ \ }\frac{\overline{X} - \mu}{S/\sqrt{n}}\sim t(n - 1)  S/n​X−μ​∼t(n−1)

4.重要公式与结论

(1) 对于χ2∼χ2(n)\chi^{2}\sim\chi^{2}(n)χ2∼χ2(n),有E(χ2(n))=n,D(χ2(n))=2n;E(\chi^{2}(n)) = n,D(\chi^{2}(n)) = 2n;E(χ2(n))=n,D(χ2(n))=2n;

(2) 对于T∼t(n)T\sim t(n)T∼t(n),有E(T)=0,D(T)=nn−2(n>2)E(T) = 0,D(T) = \frac{n}{n - 2}(n > 2)E(T)=0,D(T)=n−2n​(n>2);

(3) 对于F ~F(m,n)F\tilde{\ }F(m,n)F ~F(m,n),有 1F∼F(n,m),Fa/2(m,n)=1F1−a/2(n,m);\frac{1}{F}\sim F(n,m),F_{a/2}(m,n) = \frac{1}{F_{1 - a/2}(n,m)};F1​∼F(n,m),Fa/2​(m,n)=F1−a/2​(n,m)1​;

(4) 对于任意总体XXX,有 E(X‾)=E(X),E(S2)=D(X),D(X‾)=D(X)nE(\overline{X}) = E(X),E(S^{2}) = D(X),D(\overline{X}) = \frac{D(X)}{n}E(X)=E(X),E(S2)=D(X),D(X)=nD(X)​

了解了基本的数学知识后,下面就进入到编程环节了,下一课将会给大家介绍Python的入门指南,让零基础的同学也能迅速学会上手Python

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