Math 2568 Summer 2021
Final Exam
Testing guidelines
•) Show all of your work. Correct answers with no supporting work may receive no credit. •)
Circle or highlight your final answers.
•) You may use your textbook and class notes.
Problem Maximum Recorded
Number Point Value Point Value
Problem 1 10
Problem 2 10
Problem 3 10
Problem 4 10
Problem 5 10
Problem 6 10
Problem 7 12
Problem 8 16
Problem 9 11
Problem 10 15
Problem 11 16
Problem 12 10
Problem 13 10
Problem 14 10
Problem 15 10
Problem 16 10
Problem 17 10
Problem 18 10
Total 200
Problem 1 Let A =
3 1 5 7 −42 11 8 −3 12
13 9 10 −6 8
.
a) [4 pts] Find a basis for N(A) in rational format.
b) [3 pts] Find a particular solution to the matrix equation A ∗ x =
5−2
14
.
c) [3 pts] Use your answers in a), b) and the Superposition Principle to express the general
solution in vector form to the matrix equation in b).
Problem 2 [10 pts] For a fixed matrices B,C ∈ R2×2, let W = {A ∈ R2×2 | A ∗ B = 2A ∗ C}.
Determine if W is a subspace of R2×2 (either prove that it is, or show via specific counterexample
that it isn’t).
Problem 3 Let L : R4 → R3 be given by L
x1
x2
x3
x4
=
(3×1 − 4×2 + 11×4)(15×2 + 9×3 − 21×4)
(−6×1 + 9×2 + 4×3 − 5×4)
.
a) [4 pts] Show that L is a linear transformation, and find the matrix representation A of L with
respect to the standard bases for R4 and R3.
b) [3 pts] Use part a) to find a basis for ker(L).
c) [3 pts] Use part a) for find a basis for im(L).
Problem 4 A set of vectors is given by S = {v1,v2,v3} in R3 where
ev1 =
5−4
7
, ev2 =
23
0
, ev3 =
11−6
10
a) [3 pts] Show that S is a basis for R3.
b) [4 pts] Using the above coordinate vectors, find the base transition matrix eTS from the basis
S to the standard basis e. Then compute the base transition matrix STe from the standard basis
e to the basis S.
c) [3 pts] If ev =
212
−4
, compute Sv (the coordinate vector of v with respect to the basis S).
Use this to express v as a linear combination of the vectors in S.
Problem 5 Six data points are given by (−4, 2), (−1, 5), (0, 10), (2, 7), (6, 13), and (8, 9).
a) [3 pts] Find the least-squares fit by a linear function.
b) [3 pts] Find the least-squares fit by a quadratic function.
b) [4 pts] Find the smallest degree polynomial which fits the points exactly.
Problem 6 A bilinear pairing on R2 is given on basis vectors by
< e1, e1 >= 13; < e1, e2 >=< e2, e1 >= 7; < e2, e2 >= 26
a) [3 pts] Find the matrix representation of the pairing.
b) [4 pts] Explain why the bilinear pairing defines an inner product.
c) [3 pts] If v = [5 − 3]T , find a non-zero vector w with < v,w >= 0
Problem 7 Let A =
4 5 −1 3 7
5 0 2 1 2
−1 2 9 −4 1
3 1 −4 2 0
7 2 1 0 10
a) [4 pts] Using the [V,D] command in MATLAB with rational format, find a diagonal matrix
D and a matrix V of maximal rank satisfying the matrix equation A ∗ V = V ∗ D. Is A
real-diagonalizable?
b) [4 pts] Write down the eigenvalues of A. For each eigenvalue, find a basis for the corresponding
eigenspace of A.
c) [4 pts] Define a pairing on R5 by < v,w >= vT ∗A ∗w. Show that this pairing is a bilinear
symmetric pairing on R5. Is the pairing an inner product? (you should justify your answer
either way)
Problem 8 Let P4 be the space of polynomials of degree less than 4 with real coefficients.
Define L : P4 → P4 by
L(p(x)) = 5x2p′′′(x)− (3x + 2)p′′(x) + 7p′(x)
a) [5 pts] Find the matrix representing L with respect to the standard basis S = {1, x, x2, x3}
of P4. Explain how this can be used to prove directly that L is a linear transformation.
b) [4 pts] Let S′ = {(4 + 3x), (2− x3), (1 + 5x− x2), (x + x3)}. Show that S′ is a basis for P4.
c) [4 pts] Compute the base transition matrix S′TS .
d) [3 pts] Use a) and c) to compute S′LS′ , the matrix representative of L with respect to the
basis S′.
Problem 9 Let u1 =
3
4
−2
5
1
, u2 =
2
−1
0
5
3
, u3 =
7
−2
9
1
7
. Also let v =
−4
10
−6
0
11
.
a) [4 pts] Compute prW (v) where W = Span{u1,u2,u3} ⊂ R5.
b) [4 pts] Compute prW⊥(v) where W
⊥ denotes the orthogonal complement of W in R5.
c) [3 pts] Compute the distance between v and W .
Problem 10 Let w1 =
5
1
3
2
9
−6
, w2 =
3
−2
−5
0
−7
3
, w3 =
−3
0
7
11
1
2
. Let W = Span{w1,w2,w3} ⊂ R
6.
a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W . You should
explicitly show each step of your calculation.
b) [5 pts] Let v =
10
−7
11
4
−1
5
Compute the projection prW (v) of v onto the subspace W using the
orthogonal basis in a).
c) [4 pts] Use the computation in b) to compute the distance between v and W .
Problem 11 Let A =
3 16 16 0
−1 −5 −2 15
0 0 −3 −15
0 0 2 8
.
a) [3 pts] Compute the characteristic polynomial of A and find its roots.
b) [4 pts] For each eigenvalue of A find a basis for the corresponding eigenspace.
c) [3 pts] Determine if A is defective. Justify your answer.
d) [6 pts] If A is defective, determine the defective eigenvalue or eigenvalues, and find a Jordan
chain (or set of Jordan chains) in the corresponding generalized eigenspace that provides a
canonical basis for that space.
Problem 12 Let B be the matrix given by
A =
4 0 −2a b a
b (a + b) 2b
where a and b are indeterminates.
a) [6 pts] Using row operations that exist for all values of a or b, together with cofactor expansion,
compute the determinant of A expressed as a function of a and b.
b) [4 pts] Use this to determine a relation between a and b that provides necessary and sufficient
conditions for the matrix A to be singular (your relation should be an equation involving a and
b).
For the following six questions, indicate whether the following statements are true or false. In
each case give a reason for your answer.
Problem 13 [10 pts] If L : V →W is a linear transformation of vector spaces and U ⊂W is a
subspace of W , then {v ∈ V | L(v) ∈ U} ⊂ V is a subspace of V .
Problem 14 [10 pts] The set {A ∈ R2×2 | A is nonsingular} is a subspace of R2×2.
Problem 15 [10 pts] If A,B are two n × n matrices, then A is similar to B if and only if
pA(t) = pB(t).
Problem 16 [10 pts] For an n× n matrix A, pA(t) = t · q(t) for some polynomial q(t) precisely
when Det(A) = 0.
Problem 17 [10 pts] If W ⊂ Rn is a subspace and v ∈ Rn, then prW (v) is the least-squares
approximation to v by a vector in W except when prW (v) = 0.
Problem 18 [10 pts] If A is a real n× n matrix, then the pairing defined by
< v,w >:= vT ∗AT ∗A ∗w
is an inner product on Rn if and only if A is invertible.