data留学生代做、代写R程序设计、R编程语言调试

日期：2020-03-16 09:09

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 inverses

of nonzero elements.

(c) Verify that matrix multiplication is commutative when restricted to A (which

completes the verification that A forms a field).

(d) Verify that B is closed under taking matrix inverses of nonzero elements, but

B 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 no

need 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 rules

T(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 R

is the rotation of R2 by ✓ radians anticlockwise about the origin. (You do not

need to verify these facts.)

(a) Write down the rules for T 1

and R1

with brief justifications.

Let C be some general curve in R2 defined by the equation

f(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 by

the equation

f(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 equation

f(x + x0, y + y0)=0

and R(C) by the equation

f(x cos ✓ + y sin ✓, x

sin ✓ + y cos ✓)=0 .

(d) Show that the equation

f(xx0) cos ✓ + (yy0) sin ✓ + x0, (xx0) sin ✓ + (yy0) cos ✓ + y0= 0

defines the curve T 1(R(T(C))),

that is, the curve that results as the image of C after first applying T, then

applying R and finally applying T 1.

(e) Let a, b, c 2 R. Use your answer to part (d), or otherwise, to deduce that, if

we rotate the line with equation

ax + by = c

about the point (x0, y0) by ✓ radians anticlockwise, then we obtain the line

with equation

(a cos ✓bsin ✓)x + (a sin ✓ + b cos ✓)y= c + ax0(cos ✓1) + y0 sin ✓

bx0 sin ✓ + y0(1cos ✓).