Test #1
Solving a Linear System (10 points)
Consider the homogeneous linear system
x1
2x1
2x2 + x3 = 0
3x2 x3 = 0
(a) Write the linear system as a matrix equation A~x = ~b. (2 points)
(b) Write the linear system as a vector equation x1~a1 + x2~a2 + x3~a3 = ~b. (2 points)
(c) Compute the reduced echelon form of the augmented matrix of the linear system. (3
points)
(d) Compute the solution set of the linear system and write the solution set as the span of
a vector. (3 points)
Inventions (10 points)
(a) Invent a linear system of two equations in two variables that has infinitely many solutions.
(2 points)
2 3
2 3
1
0
(b) Invent a vector ~v with no zero entries that is a linear combination of 415 and 415. (2
0
1
points)
(c) Invent a 2 ⇥ 4 matrix A whose columns do not span R2 . (2 points)
2
3
1 0 
x
(d) Invent a vector ~b in R3 so that 40 15 1 = ~b has no solution. (2 points)
x2
0 0
2 3

1
1
(e) Invent a matrix A so that 425 is a solution of A~x =
. (2 points)
4
3
Solving Another Linear System (10 points)
2
0
6 6
Consider the linear system 6
41
1
3
2
3
0 
0
6
7
127
7 x1 = 6 127.
4 2 5
2 5 x2
2
2
(a) Compute the reduced echelon form of the augmented matrix of the linear system. (3
points)
1
(b) Write the solution set of the system in parametric vector form. (3 points)
(c) Sketch the solution set in the plane R2 . (2 points)
(d) Is this solution set the span of a vector? Why or why not? (2 points)
Span (10 points)
(a) Sketch the set Span
(b) Sketch the set Span
⇢
⇢
1
3
in the plane R2 . (2 points)

1
0
,
3
1
in R2 . (2 points)
(c) What can you deduce about vectors ~v and w
~ in R3 if Span{~v , w}
~ is just one point? (2
points)
(d) What can you deduce about vectors ~v and w
~ in R3 if Span{~v , w}
~ is a line? (2 points)
(e) Suppose you know that Span{~v , w}
~ is a plane in R3 . Suppose ~u in R3 is not on that
plane. What is Span{~v , w,
~ ~u}? (2 points)
Echelon Form (10 points)
2
0 1 4
4
Consider the matrix A = 0 0 1
0 0 0
0
2
0
3
3
35.
0
(a) Describe a row operation in words that would put A into reduced echelon form. (2
points)
(b) Which columns of A are the pivot columns? (2 point)
(c) In the equation A~x = ~b, which are the free variables? (2 point)
(d) What can you say about ~b in R3 if you know that A~x = ~b has no solution? (1 points)
(e) Describe the set of all ~b in R3 for which A~x = ~b has exactly one solution. (1 points)
(f) Suppose that a matrix B can be row reduced to get the echelon matrix A. What can
you deduce about the first column of B? What can you deduce about the last row of
B? (2 points)
2
Test #2
Inventions (10 points)
(a) Invent a 3 ⇥ 3 matrix A that is not invertible. (2 points)
(b) Invent 2 ⇥ 2 matrices A and B such that AB 6= BA. (2 points)
(c) Invent a 2⇥3 matrix A with no zero entries such that the linear transformation T : R3 !
R2 defined by T (~x) = A~x is not onto. (2 points)
(d) Invent a 2 ⇥ 2 matrix A such that A 6= I2 and A2 = I2 . (2 points)
(e) Invent a set of linearly independent vectors in R3 that contains as many vectors as
possible. (2 points)
Inverse Matrix (10 points)
2
3
1 0 0
Let A = 4 0 1 15.
2 0 1
⇥
⇤
(a) Let B be the inverse of A. Compute B using row reduction on A I3 . (5 points)
(b) Show B is the inverse of A by computing BA. (2 points)
2 3
1
(c) Use B to solve A~x = 4 35. (3 points)
4
Geometry (10 points)
Let A =

1
0
2
and let T : R2 ! R2 be the linear transformation defined by T (~x) = A~x.
1
⇢
1
0
(a) In one copy of R , draw Span
and Span
. In another copy of R2 , draw
0
1
what you get when T acts on all points of those two lines. (4 points)




0
1
0
1
(b) In one copy of R2 , draw the square with vertices
,
,
,
. In another copy of
0
0
1
1
R2 , draw what you get when T acts on all points of that square. (4 points)
2
⇢
(c) What geometric shape would you get if T acts on all points of a line that does not
contain the origin? (1 point)
(d) What geometric shape would you get if T acts on all points of a parallelogram that is
far from the origin? (1 point)
3
One-to-one (10 points)
2
1
4
2
Let A =
1
3
2
4 5 and let T : R2 ! R3 be the linear transformation defined by T (~x) = A~x.
2
(a) Write down a linear dependence relation among the columns of A. (2 points)
(b) Based on your answer in (a), is T one-to-one? (1 point)
(c) Invent two distinct vectors ~x and ~y in R2 such that T (~x) = T (~y ). Hint: try to use your
answer in (a). (2 points)
(d) Compute the echelon form of A. (3 points)
(e) Look at the pivots in your answer for (d). Is T one-to-one? Explain. (2 points)
Linear Transformations (10 points)
Let T : R3 ! R2 be a linear transformation such that T (~e1 ) =

02 3 1

0
1
2
@
4
5
A
1
and T
=
.
2
3
1
(a) Compute each of the following vectors if you can, or state that not enough information
is given: (3 points)
02 3 1
0
@
4
05A
(i) T
0
02 3 1
0
@
4
15A
(ii) T
0
02 3 1
1
(iii) T @415A
1
Now suppose you also know that T (~e3 ) =

0
.
5
(b) Compute all vectors you couldn’t compute in (a). (3 points)
(c) Compute a matrix A such that T (~x) = A~x for all ~x in R3 . (4 points)
4
Test #3
Inventions (10 points)
(a) Invent a matrix A such that det A =
4 and det(3A) =
36. (2 points)
(b) Invent a square matrix A such that rank(A) = 2 and det A = 0. (2 points)
(c) Invent a vector space V whose objects are matrices such that dim V = 100. (2 points)
(d) Invent a 2 ⇥ 2 matrix A such that AT 6= A and Col A = Row A. (2 points)
2 3
1
(e) Invent a basis B = {~p1 , p~2 , p~3 } of P2 such that [t2 ]B = 415. (2 points)
0
Subspaces Associated to a Matrix (10 points)
2
1
Let A = 4 0
2
6
0
12
3
3
0 5.
6
(a) Compute the reduced echelon form of A. (2 points)
(b) Compute a basis for Nul A. What is dim(Nul A)? (4 points)
(c) Write down a basis for Col A. What is dim(Col A)? (2 point)
(d) Write down a basis for Row A. What is dim(Row A)? (2 point)
Determinant (10 points)
(a) Let H be the set of all matrices A in M2⇥2 such that det A = 0. Answer each of the
following questions. You do not need to justify your answers. (4 points)
(i) Is ~0 in H?
(ii) Is H closed under vector addition?
(iii) Is H closed under scalar multiplication?
(iv) Is H a subspace of M2⇥2 ?
(b) Is the set of polynomials {1 + 2t, 3t + 3t2 , 4 t + 7t2 } a basis for P2 ? Use a coordinate
mapping to reduce the problem to computing the determinant of a 3 ⇥ 3 matrix. Show
your work! (6 points)
5
Change of Basis (10 points)
Consider the basis B =
⇢

2
2
,
1
0
of R2 .
(a) Compute the B-coordinates of each of the following vectors in R2 : (3 points)

2
(i)
1

2
(ii)
0

4
(iii)
1
(b) Write down the matrix P that converts B-coordinates to standard coordinates. MulE
B
tiply P by one of your answers in (a) to check that you get back the original vector.
E
B
(3 points)
(c) Compute the matrix P that converts standard coordinates to B-coordinates. Compute

B E
2
4
P
and P
. You should get your answers in (a). (4 points)
1
B E
B E 1
Coordinates and Linear Transformations (10 points)
Let P2
T
/
R be the linear transformation defined by T (~p) =
2

p~( 1)
.
p~(2)
(a) Compute T (1), T (t), and T (t2 ). (2 points)
Let E = {1, t, t2 } be the standard basis of P2 . Using the coordinate mapping P2
S
we can view T as a linear transformation R3
/ R2
[ ]E
/
R3 ,
defined by S(~x) = A~x.
(b) Write down the matrix A. Hint: the columns of A are your answers in (a). (2 points)
(c) Use the matrix A to check that S([t2 ]E ) = T (t2 ). (1 point)
(d) Compute a basis for Nul A. (4 points)
(e) Use your answer in (d) to write down a basis for the kernel of T . (1 point)
6
Test #4
Inventions (10 points)
(a) Invent a matrix A and a nonzero vector ~x that is not an eigenvector of A. (2 points)

1
(b) Invent a 2 ⇥ 2 matrix A such that
is a 3-eigenvector of A. (2 points)
2
(c) Invent a matrix A with all entries nonzero that has 0 as an eigenvalue. (2 points)
(d) Invent a matrix A that has a two-dimensional eigenspace. (2 points)
(e) Invent a 2 ⇥ 2 matrix A with real eigenvalues that is not diagonalizable. (2 points)
Definition of Eigenvectors (10 points)
(a) Write down the equation that defines what it means for ~x to be a -eigenvector of a
matrix A. (2 points)


7
1 7
(b) Show that
is an eigenvector of
and compute its eigenvalue . (2 points)
2
0
1
(c) Suppose ~x is a 2-eigenvector of A and a 5-eigenvector of B. Show that ~x is an eigenvector
of AB and compute its eigenvalue . (2 points)
2
3
3 0 2
(d) Compute a basis for the 6-eigenspace of the matrix A = 4 6 6 45. (4 points)
3 0 8
Diagonalization (10 points)
Let A =

3 1
.
2 2
(a) Compute the eigenvalues
1,
2
of A. (2 points)
(b) Compute independent eigenvectors ~v1 , ~v2 of A. (4 points)
(c) Write down the diagonalization A = P DP
and D is diagonal. (2 points)
1
, where the columns of P are eigenvectors
(d) Use your answer in (c) to write down a nice formula for A2015 . (2 points)
7
Matrix for the Derivative (10 points)
T
/ P2 be the linear transformation that takes the derivative. For example, T (5 +
Let P3
t2 3t3 ) = 2t 9t2 . Let E = {1, t, t2 , t3 } and E 0 = {1, t, t2 } denote the standard bases for P3
and P2 .
3t3 and the E 0 -coordinates of 2t
(a) Write down the E-coordinates of 5 + t2
points)
9t2 . (2
(b) Compute the E 0 -coordinates of T (1), T (t), T (t2 ), and T (t3 ). (4 points)
(c) Write down the 3 ⇥ 4 matrix for T relative to the bases E and E 0 . (2 points)
(d) Check your matrix is correct by multiplying it by the E-coordinates of 5 + t2
found in (a). You should get the E 0 -coordinates of 2t 9t2 . (2 points)
3t3 you
Discrete Dynamical System (10 points)
Let A be a 2 ⇥2 matrix with eigenvalues 1 = 1, 2 = 13 and corresponding eigenvectors
2
1
~v1 =
, ~v2 =
. Let B = {~v1 , ~v2 } be the eigenvector basis for R2 .
1
3
(a) Compute the change of basis matrix P . (3 points)

B
E
30
(b) Let ~x0 =
. Compute c1 and c2 such that ~x0 = c1~v1 + c2~v2 . (Hint:
50
(2 points)

c1
= P ~x0 .)
c2
B E
(c) Use your answer in (b) to write down a nice formula for Ak ~x0 . (2 points)
(d) What vector does Ak ~x0 converge to as k gets large? (1 point)
(e) Invent a vector ~y0 such that Ak ~y0 converges to ~0 as k gets large. (1 point)
(f) Invent a vector ~z0 such that Ak ~z0 converges to ~z0 as k gets large. (1 point)
8
Test #5
Inventions (10 points)
(a) Invent a unit vector in R4 with no zero entries. (2 points)
2
3
3
(b) Invent a vector with no zero entries that is orthogonal to 4 7 5. (2 points)
2
(c) Invent vectors ~x1 , ~x2 in R2 such that the orthogonal complement of Span{~x1 , ~x2 } is a
line. (2 points)
(d) Invent a symmetric 2 ⇥ 2 matrix of rank 1. (2 points)
(e) Invent a matrix that is diagonalizable but not orthogonally diagonalizable. (2 points)
Orthogonal Projection (10 points)
8 2 3 2 39
9 =
< 2
4
5
4
1 , 15 of R3 .
Consider the subspace W = Span
:
;
1
5
(a) Compute an orthogonal basis B = {~u1 , ~u2 } of W . (3 points)
2 3
9
4
(b) Let ~y = 2 5. Compute projW ~y . (4 points)
4
(c) What is the closest point in W to ~y ? (1 point)
(d) Compute the distance from ~y to W . (2 points)
Least Squares Approximation (10 points)
2
3
2 3
1 0
4
Consider the equation A~x = ~b, where A = 43 15 and ~b = 415.
0 1
0
(a) Determine whether A~x = ~b has a solution. (2 points)
(b) Compute the unique solution ~x0 of AT A~x = AT~b. (5 points)
(c) Compute kA~x0
~bk. (2 points)
(d) What can you say about ~x0 to relate it to the original equation A~x = ~b? (1 point)
9
Orthogonal Diagonalization (10 points)
Let A =

4
2
2
.
7
(a) Why can you immediately tell that A is diagonalizable? (1 point)
(b) Compute the eigenvalues of A. (3 points)
(c) Compute a basis {~v1 , ~v2 } of R2 consisting of orthonormal eigenvectors of A. (4 points)
(d) Write down the diagonalization A = P DP T , where the columns of P are orthonormal
eigenvectors and D is diagonal. (2 points)
Singular Value Decomposition (10 points)
Let
2
2
44
0

1 2
1
1 2 1
3 2 p
4 0
1/p5
5
4
8 0 = 2/ 5
0 2
0
A =
and assume that
p 32
0
2/p 5
10
5
4
0
0 1/ 5
0
1
0
an orthogonal diagonalization of AT A is given by
p
32 p
3
0 0
1/ 5 2/ 5 0
2 05 4 0p
0p 15.
0 0
2/ 5 1/ 5 0
(a) Compute the singular value decomposition A = U ⌃V T . (Hint: V and ⌃ can easily be
obtained from the given information!) (7 points)
(b) What is the rank r of A? (1 point)
(c) Compute the best rank 1 approximation of A. (2 points)
10