Linear Algebra Linear Transformation Orthogonality Problems

1
Let W be a subspace of Rn , and recall that h·, ·i is the usual dot product of vectors in Rn . Recall that we
define the orthogonal complement of W to be the set
W ⊥ = {~v ∈ Rn : h~v, w
~ i = 0 for all w
~ ∈ W} .
(a) Show that W ⊥ is a vector subspace of Rn.

 
0
1
0
−3
 
 
5
⊥


(b) If W is the span of the vectors w
~1 = 
~2 = 
 1  in R , give a basis for W .
 0  and w
−5
4
2
−6

2


1 2
3
4
5
6 7
8
9 10

Let M = 
11 12 13 14 15. You may cite Sage calculations for part (a) of this problem as part of
16 17 18 19 20
your justification, but be sure to indicate what calculation you are doing and what the output is.
(a) Find a basis for each of the four fundamental spaces of M : the nullspace of M , the column space of
M , the row space of M , and the left nullspace of M .
Problem #2 continued

1
6
We continue to use M = 
11
16
(b)

M
M

Let A =  .
 ..
2
7
12
17
M
M
..
.
3
8
13
18
…
…
..
.
4
9
14
19

5
10
.
15
20

M
M

..  be the 1000 × 2020 matrix consisting of blocks of the 4 × 5
. 
M M … M
matrix M . What is the rank of A? Justify your reasoning. (Do NOT try to do this on Sage.)
(c) What are the dimensions of the four fundamental subspaces of A: the nullspace of A, the column
space of A, the row space of A, and the left nullspace of A?
3
Read each statement carefully and determine whether each statement, as written, is TRUE or FALSE.
Then justify your conclusion either by justifying why the statement is true or giving a counterexample to
show the statement is false.
(a) If A is a 3 × 3 matrix with eigenvalues 1, 2 and 3, then A2 has eigenvalues 1, 4 and 9.
(b) If A is a 2 × 2 matrix with eigenvalues 1 and 2, and B is a 2 × 2 matrix with eigenvalues 3 and 4,
then A + B has eigenvalues 4 and 6.
4


1
Let L be the line spanned by the vector ~v = −1 in R3 and let P = L ⊥ be the plane in R3 (passing
2
through the origin) that is orthogonal to L . You may cite Sage calculations for this problem as part of
your justification, but be sure to indicate what calculation you are doing and what the output is.
(a) Find an orthonormal basis B = {q~1, ~q 2, ~q 3} of R3 such that ~q 1, ~q 2 forms an orthonormal basis for P and
~q 3 is an orthonormal basis of L .
Problem #4 continued
We keep all of the notation from the previous page.
(b) (10 points) Let
P : R3 →R3 be the
that projects a vector orthogonally onto the
 linear transformation
 
1
0
0 

plane P. Let E = ~e1 = 0 , ~e2 = 1 , ~e3 = 0 be the standard basis of R3 . Find the matrix


0
0
1
E
A = [P ]E which represents P in the standard basis, i.e. for any vector ~x in R3 , A~x is the orthogonal
projection of ~x onto P.
5

1
2
√
2
4
−

1
Let A = 
2
− 12 −
√
2
4
−√12
− 22
− 21
− 12
1
2
√
2
4
−
1
2√
−
2
4


.
(a) Show that the matrix A is orthogonal. Show all your calculations.
(b) Find an eigenvector ~v of A with eigenvalue λ = 1. Show all your calculations.
Problem #5 continued
We keep all of the notation from the previous page. You may cite Sage calculations for parts (c) and (d) of
this problem as part of your justification, but be sure to indicate what calculation you are doing and what
the output is.
(c) Let W be the orthogonal complement of the line spanned by the vector ~v that you found in part(b).
Show that if ~x belongs to W , then so does A~x. (Hint: First, show that this is true for vectors ~x in a basis
of W . Then explain why this proves that the statement is true for any vector ~x in W .)
(d) The vector A~x is the rotation of the vectors ~x in W by an angle of θ. Determine the angle θ. There are two
possible answers, depending whether you view the rotation clockwise or counterclockwise from a given
perspective, and either answer is acceptable. (Hint: For the linear transformation
LA : W → W (where LA (~x) = A~x), the matrix [LA ]B
B for any orthonormal basis will be a rotation matrix
cos θ − sin θ
R(θ) =
where θ is the angle of rotation.)
sin θ cos θ