讲解：data、R、RWeb|Database

1. Consider the following sets of real 2 ⇥ 2 matrices:(a) Verify that A and B are closed under matrix addition and taking negatives(so both A and B form abelian groups with respect to addition).(b) Verify that A is closed under matrix multiplication and taking matrix inversesof nonzero elements.(c) Verify that matrix multiplication is commutative when restricted to A (whichcompletes the verification that A forms a field).(d) Verify that B is closed under taking matrix inverses of nonzero elements, butB is not closed under matrix multiplication (so B does not become a field).(e) Let : C ! A be the map.Then clearly is a bijection (and there is noneed to verify this). Verify that(z + w) = z + w and (zw) = (z)(w) .for all z, w 2 C. (This verifies that C and A are isomorphic fields.)2. Fix x0, y0, ✓ 2 R and define bijections T,R : R2 ! R2 by the rulesT(x, y)=(xx0, yy0)andR(x, y)=(x cos ✓y sin ✓, x sin ✓ + y cos ✓) .Thus T is the parallel translation of R2 that takes (x0, y0) to the origin, and Ris the rotation of R2 by ✓ radians anticlockwise about the origin. (You do notneed to verify these facts.)(a) Wdata留学生作业代做、代写R程序设计作业、R编程语言作业调试 代写Web开发|代写Databaserite down the rules for T 1and R1with brief justifications.Let C be some general curve in R2 defined by the equationf(x, y)=0 ,where f(x, y) is some algebraic expression involving x and y, that is,C = { (x, y) 2 R2 | f(x, y)=0 } .(b) Verify carefully that if B : R2 ! R2 is any bijection then B(C) is defined bythe equationf(B1(x,y)) = 0 ,that is,B(C) = { (x, y) 2 R2 | f(B1(x,y)) = 0 } .(c) Deduce from parts (a) and (b) that T(C) is the curve defined by the equationf(x + x0, y + y0)=0and R(C) by the equationf(x cos ✓ + y sin ✓, xsin ✓ + y cos ✓)=0 .(d) Show that the equationf(xx0) cos ✓ + (yy0) sin ✓ + x0, (xx0) sin ✓ + (yy0) cos ✓ + y0= 0defines the curve T 1(R(T(C))),that is, the curve that results as the image of C after first applying T, thenapplying R and finally applying T 1.(e) Let a, b, c 2 R. Use your answer to part (d), or otherwise, to deduce that, ifwe rotate the line with equationax + by = cabout the point (x0, y0) by ✓ radians anticlockwise, then we obtain the linewith equation(a cos ✓bsin ✓)x + (a sin ✓ + b cos ✓)y= c + ax0(cos ✓1) + y0 sin ✓bx0 sin ✓ + y0(1cos ✓).转自：http://www.6daixie.com/contents/18/4984.html