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(math111)[2008](f)MATH111Q5~PPSpider^_10430.pdf
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MATH111 Quiz 5A Solutions
Page 1/4
November 28, 2008 by Daniel Zheng
..................................................40 mins allowed..................................................

Name: Student ID: Score:
1012 1 .123
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I.Considermatrix A ..1432=. . 1.[5]Findan orthogonal basisofNul A. Solutions. FindabasisforNulA,thenuseGram-Schmidtprocesstoorthog-TT{...}onalizethebasis. (1110)(1011),,,,,,, .
..
.
MATH111 Quiz 5A Solutions Page 2/4
2. [3] Find the distance between u = (1, 1, 1, 1) and RowA. That is |
u . projrowAu|
. (Hint: Using the fact (RowA) = NulA and your result of part (1)). Solutions. Because (RowA) = NulA, we have
projNulAu = u . projrowAu. By part (1), projNulAu = (0, 1/3, 2/3, .1/3)T . Hence the distance is
.
.12 1.T .
.2
0,,, . = .
333 3
MATH111 Quiz 5A Solutions Page 3/4
II. [4] Let T : Rn Rn be a linear transformation that preserves lengths. That is, |
T x|
= |
x|
for all x Rn . Show that
1.
T also preserves orthogonality. That is, T x T y = 0 whenever x y = 0.

2.
The standard matrix of T is an orthogonal matrix.


Proof.
1. Suppose x y = 0, by Pythagorean Theorem, .x.2 + .y.2 = .x + y.2 . Since T preserves lengths,
.T x.2 + .T y.2 = .x.2 + .y.2 = .T (x + y).2 = .T x + T y.2 .
Again by Pythagorean Theorem, T x T y = 0.
2. {T e1, ,T en} is an orthonormal set, because T preserves both orthogonality and lengths. A square matrix with orthonormal columns is an orthogonal matrix.
MATH111 Quiz 5A Solutions Page 4/4
III. Determine whether the following statements are true or not. If true, justify your answer; if false, give a counterexample.
1.
[2] If two vectors are orthogonal, they are linearly independent.

2.
[2] If a square matrix has orthogonal columns, it also has orthogonal rows.

3.
[2] projspan{u, v}x = projspan{u}x + projspan{v}x.


Solutions.
1.
False. Consider u = (0, 0)T , v = (1, 0)T .
10


2.
False. Consider A =.
10


3.
False. Consider u = v.


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