Homework 07 Solution

LA(~x) = A~x, with A =

a

c

b

d

, is linear.

2. Determine whether each of the following functions is linear or not. Explain your reasoning.

2. 1. T : R ! R, T (x) = x².

2. 2. T : R ! R, T (x) = x + 3.

(c) T : R2	!R2,T	x2		=	x1		+ 2x2
		x1					3x1
(d) T : R2	!R2,T	x2		=		2x2		1
		x1					x1

3. Assume that T : R2	! R2	is a linear transformation. Let ~e1 =	1	and ~e2	=	0	. Draw the image of the
			0			1

”half-shaded unit square” (shown below) under the given transformation T , and find the matrix A such that T = L_A.

!A!

?!

1. T stretches by a factor of 2 in the x-direction and by a factor of 3 in the y-direction.

2. T is a reflection across the line y = x.

(c) T is a rotation (about the origin) through

=4 radians.

(d) T is a vertical shear that maps ~e1 into ~e1

~e2 but leaves the vector ~e2 unchanged.

4. For any given m n matrix A, we are going to use the notation L_A to denote the linear transformation that A defines, i.e., L_A : Rⁿ ! R^m : L_A(~x) = A~x. For each given matrix, answer the following questions.

D =	20	1	0 3	E =	20	03	F = 0	3	0
	3	0	0		4	0		0
	0	0	1=2		0	2
	4		5		4	5	2		1

1. Rewrite L_D : Rⁿ ! R^m with correct numbers for m and n filled in for each matrix. Repeat for L_E and L_F .

2. Find some way to explain in words and/or graphically what this transformation does in taking vec-tors from Rⁿ to R^m. You might find it helpful to try out a few input vectors and see what their image is under the transformation.

3. Is this transformation one-to-one? (Hint: Review problem#6 of Homework 06.)

1. 1. If so, explain which properties of the matrix make the transformation one-to-one.