Math 2270, Fall 2014
Instructor: Thomas Goller
14 December 2014
All Quiz Problems
Quiz #1


1
1
(1) Find two di↵erent linear combinations of the three vectors ~u =
and ~v =
and
0
1


0
1
w
~=
that produce ~b =
. (4 points)
1
1
2 3
1
(2) Find two di↵erent vectors ~v and w
~ that are perpendicular to 4 0 5 and to each other.
1
(4 points)
2 3
2 3
2 3
1
1
0
(3) Let ~u = 415 and ~v = 405 and w
~ = 4 2 5. Compute the linear combination
1
0
2
~
4~u 3~v + w
~ = b and write it as a matrix-vector multiplication A~x = ~b. (4 points)
Quiz #2
(1) ⇥Compute
⇤ the inverse of A using the Gauss-Jordan method starting with the block matrix
A I . Show your work!
2
3
1 0 5
A = 44 2 20 5
(8 points)
0 4
5
(2) Write the following problem in a 2 by 2 matrix form A~x = ~b:
“X is three times as old as Y and their ages add to 72.”
You do not need to solve the problem! (4 points)
Quiz #3
(1) True or false (you don’t need to justify your answer):
(a) If A and B are symmetric then AB is symmetric. (1 point)
(b) The line in R2 with equation x + y = 0 is a subspace. (1 point)
2
3
1
2
4 5 is a plane in R3 . (1 point)
(c) The column space of 4 2
5
10
Math 2270, Fall 2014
Instructor: Thomas Goller
14 December 2014
(d) (AB)T equals AT B T . (1 point)
(2) Let P be the plane in R3 with equation 2x
check that their sum is not in P . (This shows
2
1
(3) Find the A = LU factorization for A = 4 0
1
y + z = 2. Find two vectors in P and
that P is not a subspace!) (4 points)
3
2 3
5 35. (4 points)
2 4
(4) Optional problem if you finish the other parts early: Find all 2 ⇥ 2 symmetric matrices
A with integer entries that satisfy A2 = I. (Hint: there are six.) (0 points)
Quiz #4
(1) Find bases and dimensions for the four subspaces associated with A =
points)
(2) Show that the following vectors are dependent:
2 3
2 3
2 3
1
0
1
4
5
4
5
4
~v1 = 1
~v2 = 2
~v3 = 0 5 .
0
2
1
(3) Optional problem: Let R =

cos ✓
sin ✓

1 2 3
. (9
0 0 0
(3 points)
sin ✓
. Check that the columns of R are perpencos ✓
dicular unit vectors. Thinking of R as a function R2 R / R2 , how does R act on vectors
in the plane? (If you’re stuck,first try to answer this question for ✓ = ⇡2 by computing
1
0
cos ✓ sin ✓
R
and R
.) How does
act on vectors in the plane, and how can
0
1
sin ✓ cos ✓
you deduce this from your answer for R? (0 points)
Quiz #5
(1) Write down three equations for the line b = C + Dt to go through b = 1 att = 1,
C
b = 1 at t = 0, and b = 3 at t = 1. Find the least squares solution x
b=
and
D
draw the closest line. (Hint: write your equations as A~x = ~b, which has no solution, so
instead solve AT A x
b = AT~b for the best approximation x
b.) (8 points)
2 3
2 3
1
2
627
627
7
6 7
(2) Find orthonormal vectors ~q1 and ~q2 in the plane spanned by 6
4 0 5 and 425. (4 points)
1
0
Math 2270, Fall 2014
Instructor: Thomas Goller
14 December 2014
(3) Optional problem: Draw a picture, write down a riddle, or tell me something you find
interesting that I probably don’t know. (0 points)
Quiz #6
All three problems below are about the matrix A =
(1) Find the eigenvalues
(2) Check that A~x1 =
1,
x1
1~
(3) Factor A into A = S⇤S
2
2 3
.
4 6
and eigenvectors ~x1 , ~x2 for A. (6 points)
and A~x2 =
1

x2 .
2~
(2 points)
. (4 points)
(4) Optional problem: Surprise me! (0 points)
Quiz #7
(1) If C = F 1 AF and also C = G 1 BG, what matrix M gives B = M 1 AM ? (You must
show your work to receive full credit!) (4 points)




✓ ◆
1
1
2
2
5
(2) Suppose a linear T transforms
to
and
to
. Find T
. (You
1
3
3
1
0
must show your work to receive full credit!) (4 points)
(3) For each of the following functions R2
have to show any work.) (4 points)
✓ ◆ 
v1
0
(a) T
=
.
v2
0
✓ ◆ 
v1
2v1
(b) T
=
.
v2
2v1
✓ ◆ 
v1
sin v2
(c) T
=
.
v2
v1
✓ ◆ 
v1
v
1
(d) T
= 1
.
v2
v2
T
/
R2 , is T linear or not linear? (You don’t
(4) (Optional fun art/geometry/creative writing problem)
 Consider the linear transforma2 1
tion R2 T / R2 defined by T (~v ) = A~v , where A =
. (T acts by a combination
0 3
of vertical and horizontal stretching and horizontal shearing.) Invent a parallelogram
in R2 that does not have the origin as one of its vertices, and check that the action of
T takes that parallelogram to another parallelogram. Sketch both parallelograms, give
them names, and write a poem about them (“O par’llel Peter, how perfect doth thine
angles be; far sweeter than sharp-cornered Cassidy!”) (0 points)