MATH 043 FINAL PRACTICE PROBLEMS (FROM OLD EXAM AND ME2) QUESTIONS

MATH 043 FINAL PRACTICE PROBLEMS (FROM OLD
EXAM AND ME2) QUESTIONS


D - Answers - Let A be a real n × n matrix. Which of the following statements is NOT
equivalent to the others?

(A) There exists a unique solution to the system of linear equations Ax = b, for each
fixed vector b ∈ Rn .
(B) The rank of A is n.
(C) The inverse matrix A−1 exists.
(D) The determinant of A is det(A) = 0.

D - Answers - Let A be a real n × n matrix. Which of the following statements is NOT
true?

(A) A + AT must be a symmetric matrix.
(B) AT − A must be a skew-symmetric matrix.
(C) AAT + ATA must be a symmetric matrix.
(D) ATA − AAT must be a skew-symmetric matrix.

D - Answers - S = {1, x, x2 , x3 , · · · · · · , x10}.
Which of the following statements about S is NOT true?

(A) S is linearly independent
(B) S is a basis of the vector space P10(R)
(C) S is a spanning set of the vector space P10(R)
(D) S is a basis of the vector space C(R)

D - Answers - Which of the following statements is NOT true?

(A) dim(R n ) = n
(B) dim[Mn(R)] = n 2
(C) dim[Pn(R)] = n + 1
(D) dim[C(R)] = n

C - Answers - Let T : V → W be a linear transformation. Which of the following
statements is NOT true?

(A) The kernel Ker(T) is a subspace of the vector space V.
(B) The range Rng(T) is a subspace of the vector space W.
(C) The inverse linear transformation T −1 exists if T is one-to-one or onto.
(D) The sum of the dimension of the kernel and the dimension of the range is equal to
the dimension of the vector space V. That is,

, dim[ Ker(T) ] + dim[ Rng(T) ] = dim V.

D - Answers - Let Rn → Rn be a linear transformation, given by T(x) = Ax, x ∈ R n ,
where A is a real symmetric matrix. All eigenvalues are positive. Which of the following
statements is NOT true?

(A) The linear transformation T is one-to-one.
(B) The linear transformation T is onto.
(C) The inverse linear transformation T −1 exists.
(D) The determinant det(A) = 0.

A - Answers - Let T : P3(R) → P3(R) be a linear transformation, given by P(a + bx + cx2
+ dx3 ) = a + bx + cx2 + dx3 . Which of the following statements is true?

(A) Ker (T) = {0}
(B) Ker (T) = span {1}
(C) Ker (T) = span {1, x}
(D) Ker (T) = span {1, x, x2 }

B - Answers - Consider the system of linear equations Ax = b, where A is an m × n real
matrix, b ∈ Rm is real vector. Let x0 represent a least square solution of the system.
Which of the following statements is true?

(A) Ax0 = b
(B) A TAx0 = A Tb
(C) AATx0 = A Tb
(D) IIAx0 − bII = max x∈Rn IIAx − bII

A - Answers - Let A be a real m × n matrix. Which of the following statements is true
about the matrix ATA?
(A) All eigenvalues of ATA are real.
(B) Some eigenvalues of ATA are real.
(C) Some eigenvalues of ATA may be complex.
(D) There may be pure imaginary eigenvalues to ATA.

D - Answers - Let A be a real m×n matrix. Let λ and µ be real distinct eigenvalues of
ATA, let ξ and η be eigenvectors of ATA corresponding to the eigenvalues λ and µ,
respectively, that is,
A TAξ = λξ, ATAη = µη.
Which of the following statements is true?

(A) ξ = η
(B) IIξII > IIηII
(C) IIξII < IIηII
(D) ξ ⊥ η