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(math113)[2010](s)hw~cwyipac^_23436.pdf
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Homework 2

Due Date: Mar 11, 2011
1. Find the value(s) of h for which the following vectors are linearly dependent. Justify
the answer. ( 1.7 Exercise 12)
..
2
.4
1
..
,
..
6
7
.3
..

,
..
8
h
4
..
.

2. Let A =
. .
. . .
. ..
and b = .
1 .2 1
3 .4 5
0 1 1
.3 5 .4

1
9
3
.6
. .
.

. If the linear transformation T is de.ned by
T (x)= Ax. Find a vector x whose image under T is b, and determine whether x is unique. ( 1.8 Exercise 6)
3. Let A =
.. ..
1 .4 7 .5
0 1 .4 3
2 .6 6 4

. Find all x in R4 that are mapped into the zero vector

by the transformation x . Ax.( 1.8 Exercise 9)
In the Exercise 4 and 5, assume that T is a linear transformation. Find the standard matrix of T .
4. T : R2 R2 rotates points(about the origin) through ./4 radians(clockwise). [Hint:


T (e1) = (1/ 2, .1/ 2).] ( 1.9 Exercise 4)
5.
T : R2 R2 .rst re.ects points through the horizontal x1-axis and then re.ects points through the line x1 = x2 ( 1.9 Exercise 8).

6.
Compute the product AB in two ways: (a) by the de.nition, where Ab1 and Ab2 are computed separately and by (b) by the row-column rule for computing AB ( 2.1 Exercise 5).


A =

..
.12
54
2 .3

..
3 .2

,B =
.21
. 86

7. Find the inverse of the matrix A =
and use the inverse to solve the system54

8x1 +6x2 =2 5x1 +4x2 = .1
8. Find the standard matrix A of the matrix transformation T : R2 R3 that satis.es
.1
= 3

.1
1

.2
=

.2
0
3

.T
2 1