MIT18.06学习笔记 - Lecture 1: The geometry of linear equations

这个系列文章是我重温Gilbert老爷子的线性代数在线课程的学习笔记。
Course Name：MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章节内容: 2.1

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课程提纲
1. N Linear Equations with N Unknowns
2. Row Picture
3. Column Picture
4. Matrix Form

课程重点

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Two Linear Equations with Two Unknowns

The central problem of linear algebra is to solve a system of equations. Those equations are linear, which means that the unknowns are only multiplied by numbers:

Row Picture

The row picture shows two lines meeting at a single point (the
solution)

Column Picture

The column picture combines the column vectors on the left side to
produce the vector b on the right side.

if we separate the original system into its columns instead of its rows, we get a vector equation:

This has two column vectors on the left side. The problem is to find the combination of those vectors that equals the vector on the right.

The left side of the vector equation is a linear combination of the column vectors.

Matrix Form

The row picture deals with the two rows of A. The column picture combines the columns:

Three equations in Three Unknowns

The row picture shows three planes meeting at a single point:

The column picture combines three columns to produce (6,4,2):

Matrix form and two ways to see multiply A and x:

This book sees Ax A x as a combination of the columns of A.

TODO
multiplication in numpy