linear algebra question bank - binghamton university

LINEAR ALGEBRA QUESTION BANK

(1) (12 points total) Circle True or False:TRUE / FALSE: If A is any n×n matrix, and In is the n×n identity

matrix, then InA = AIn = A.

TRUE / FALSE: If A,B are n×n matrices, then the inverse of AB isA−1B−1.

TRUE / FALSE: If A,B are n×n matrices, then (A+B)−1 = A−1 +B−1. (Hint: is this true for numbers?)

TRUE / FALSE: The Reduced Row Echelon Form of a matrix is unique.

TRUE / FALSE: IfA andB are both 2×3 matrices, then their productAB is defined.

TRUE / FALSE: If A and B are both 2×3 matrices, then the productABT is defined.

(2) True or false: If a system of equations has more than one solution, it hasinfinitely many solutions.

(a) True(b) False

(3) True or false: If a system of equations is consistent, then it cannot haveany free variables.

(a) True(b) False

(4) True or false: Let A be a 2× 3 matrix. Then Nul(A) is a subspace of R2.

(a) True(b) False

(5) True or false: Let A be a 2× 3 matrix. Then Col(A) is a subspace of R2.

(a) True(b) False

1

2 LINEAR ALGEBRA QUESTION BANK

(6) True or false: If V is a vector space of dimension d, and {v1, . . . ,vd} are ddifferent vectors in V , then they must form a basis.

(a) True(b) False

(7) True or false: If V is a subspace of Rn, then every basis for V must havethe same number of vectors.

(a) True(b) False

(8) True or false: If V is a vector space of dimension d, and {v1, . . . ,vd} are dlinearly independent vectors in V , then they must span V .

(a) True(b) False

(9) What is the dimension of the null space Nul(A) of A =

2 3 1 −1 00 0 4 2 00 0 0 0 0

?

A. 1B. 2

C. 3D. 5

(10) What is the dimension of the column space Col(A) ofA =

2 3 1 −1 00 0 4 2 00 0 0 0 0

?

A. 1

B. 2C. 3D. 5

(11) What is the dimension of the left null space Nul(AT ) ofA =

2 3 1 −1 00 0 4 2 00 0 0 0 0

?

A. 1B. 2C. 3D. 5

LINEAR ALGEBRA QUESTION BANK 3

(12) What is the dimension of the row space Col(AT ) ofA =

2 3 1 −1 00 0 4 2 00 0 0 0 0

?

A. 1

B. 2C. 3D. 5


For questions 5 and 6: Suppose

A =

1 −3 1 −20 1 −1 11 −1 −1 0

and its reduced echelon form is

U =

1 0 −2 10 1 −1 10 0 0 0

(13) Which of these is a basis for Col(A)?

A.

1

01

,−3

1−1

, 1−1−1

,−2

10

B.

1

00

,0

10

C.

1

01

,−3

1−1

D.

1

01

, 1−1−1

,−2

10

(14) Which of these is a basis for Col(AT )?

A. Span

1−31−2

,

01−11

,

1−1−10

B. Span

10−21

,

01−11

C.

1−31−2

,

01−11

,

1−1−10

D.

10−21

,

01−11


(15) The matrix for a 90◦ counterclockwise rotation in the x-y plane is

A.

[0 11 0

]B.

[0 −1−1 0

]

C.

[0 −11 0

]D.

[0 1−1 0

]

(16) Let L be the linear transformation from P2 to P2 given by

L(p(t)) = 2p′(t) + 3p(t)

and let B = {1, t, t2} be the standard basis for P2. Then the coordinatematrix A representing L with input and output basis B is:

A.

3 0 02 3 00 4 3

B.

2 3 00 2 60 0 2

C.

3 2 00 3 40 0 3

D.

2 0 03 2 00 6 2


(17) For every m× n matrix A, the orthogonal complement of Col(A) in Rm isNul(A).A. TrueB. False

(18) For every m × n matrix A, the sum of the dimensions of Nul(AT ) andCol(A) is equal to m.

A. TrueB. False

(19) If V is a 6-dimensional vector space, and v1, . . . ,vm is a basis for V , thenm must be equal to 6.

A. TrueB. False

(20) If V is a 6-dimensional vector space, and v1, . . . ,v6 are six vectors in V ,then they must form a basis of V .A. TrueB. False

(21) If V is a 6-dimensional subspace of R10, then the orthogonal complementV ⊥ must be 4-dimensional.A. TrueB. False

(22) If V is a 3-dimensional subspace of R7, and v1, v2, and v3 are three linearlyindependent vectors in V , then they also span V .

A. TrueB. False

(23) If V and W are subspaces of Rn, and W⊥ = V , then V ⊥ = W .

A. TrueB. False


(24) Suppose A = [a1 . . .a4] and B = [b1 . . .b4] are two 4× 4 matrices so that

AB =

1 1 2 31 2 4 61 3 6 91 4 8 12

.What is Ab2? That is, what is A times the second column of B?

(a)

1246

(b)

1234

(c) Not enough information to tell.

(25) Let

A =

0 1 10 5 26 1 2

.For which permutation matrix P does PA have an LU decomposition?

(a) P =

1 0 00 1 00 0 1

(b) P =

0 1 01 0 00 0 1

(c) P =

0 0 10 1 01 0 0

(26) Suppose A is a matrix with LU decomposition:

A =

[1 02 1

] [1 20 1

]

If b =

[11

], the LU method for Ax = b gives

(a) c =

[31

], x =

[73

].

(b) c =

[−11

], x =

[3−1

].

(c) c =

[13

], x =

[73

].

(d) c =

[1−1

], x =

[3−1

].

(27) What is the inverse of the matrix

A =

1 0 02 1 00 0 1

?


(a) A−1 =

1 −2 00 1 00 0 1

(b) A−1 =

1 0 0−2 1 00 0 1

(c) A−1 =

1 0 00 1 00 −2 1

(28) Suppose A is a 3× 3 matrix so that

A

142

=

100

, A

310

=

010

, and A

217

=

001

.

What is the first column of A−1?

(a)

132

(b)

142

(c)

310

(d)

217

(e) Not enough information to tell

(29) Suppose A and B are invertible 3× 3 matrices, with inverses

A−1 =

1 0 0−2 1 00 0 1

and B−1 =

1 0 00 1 00 5 1

What is (AB)−1?

(a)

1 0 02 1 00 −5 1

(b)

1 0 02 1 0−10 −5 1

(c)

1 0 0−2 1 00 5 1

(d)

1 0 0−2 1 0−10 5 1

(30) Which of the following are subspaces of P2, the vector space of polynomialswith degree at most 2:• W1 =

{a0 + a1t+ a2t

2 : a0 = 1, and a1, a2 ∈ R}

• W2 ={a0 + a1t+ a2t

2 : a1 = 1, and a0, a2 ∈ R}

• W3 ={a0 + a1t+ a2t

2 : a1 = 0, and a0, a2 ∈ R}

• W4 = {at+ b(t− 1) : a, b ∈ R}

(a) W3 only(b) W4 only(c) W3 and W4 only(d) W1 and W2 only(e) All four are subspaces

(31) Which of the following are subspaces of the indicated vector space?

• W1 =

abc

: a− 2b = c, 4a+ 2c = 1

⊆ R3


• W2 =

a− bc

a+ ca− 2b− c

: a, b, c ∈ R

⊆ R4

• W3 =

{[ab

]: a · b ≥ 0

}⊆ R2

• W4 =

{[ab

]: a2 + b2 ≤ 1

}⊆ R2

(a) W2 only(b) W1 and W2 only(c) W2 and W3 only(d) All four are subspaces

(32) Suppose

A =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

and B =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

are 3× 4 matrices, and b is a vector in both Col(A) and Col(B). Supposealso that

Nul(A) = {0} and Nul(B) = span

1010

Which of the following is true?

(a) Ax = b has a unique solution, but Bx = b does not.(b) Bx = b has a unique solution, but Ax = b does not.(c) Both Ax = b and Ax = b have unique solutions.(d) Neither Ax = b nor Ax = b have unique solutions.

(33) Let

A =

[1 3−2 −6

]Which of the following are in Col(A)?

• v1 =

[00

]• v2 =

[4−12

]• v3 =

[10

]• v4 =

[1−2

](a) v4 only(b) v1 and v4 only


(c) v2 and v4 only(d) v1, v2 and v4 only(e) v1, v2, v3 and v4

(34) Which of the following sets of vectors are linearly independent?

• A =

4−26

, 6−39

• B =

{[4−2

],

[60

],

[13

],

[−21

]}• C =

4−26

,6

21

,0

00

• D =

−1042

,

5−5010

,−113−11

(a) None of them are linearly independent(b) D only(c) A and D only(d) B and D only(e) C and D only

(35) True or false: If the columns of a matrix A are linearly independent, thenthe rows of A must also be linearly independent.

(a) True(b) False

(36) True or false: The dimension of Nul(A) must be equal to the number ofzero rows at the bottom of an echelon form of A.

(a) True(b) False

(37) True or false: For every matrix A, with echelon form U , the row spaceCol(AT ) must be equal to the row space Col(UT ).

(a) True(b) False

(38) True or false: If the columns of a matrix A are linearly independent, thenNul(A) must be {0}.

(a) True


(b) False

(39) True or false: For every matrix A, the column space Col(A) and null spaceNul(A) are orthogonal complements.

(a) True(b) False

(40) True or false: For every matrix A, the row space Col(AT ) and the nullspace Nul(A) are orthogonal complements.

(a) True(b) False

(41) True or false: Every orthonormal basis is an orthogonal basis.

(a) True(b) False

(42) True or false: If B = {v1, . . . ,vn} is any basis for Rn, and w is anothervector, then the projections of w onto each of the vectors v1, . . . ,vn mustsum back to w.

(a) True(b) False

(43) True of false: If A and B are two bases for Rn, and IBA is the change ofbasis matrix for input basis A and output basis B, then

vB = IBAvA

for any vector v ∈ Rn.

(a) True(b) False

(44) True or False: If a square matrix A has an eigenbasis, then A must beinvertible.

(a) True(b) False

(45) True or False: For all square matrices A and B, det(A + B) = det(A) +det(B).

(a) True


(b) False

(46) True or False: If Q is an orthogonal matrix, then det(Q) must be equal to1.

(a) True(b) False

(47) True or False: If A has an orthogonal basis of eigenvectors, then A mustbe symmetric.

(a) True(b) False

(48) True or False: In a discrete dynamical system with transition matrix A, ifall eigenvalues of A have absolute value smaller than 1, then lim

n→∞vn = 0

for every orbit v0, v1, v2, . . .. (You may assume that A has a basis ofeigenvectors.)

(a) True(b) False

(49) The determinant of the matrix A =

1 0 a 00 1 b 00 0 c 00 0 d 1

is:

A. abcdB. aC. bD. −cE. c

(50) The determinant of A =

0 0 0 10 0 1 00 1 0 01 0 0 0

is:

A. −1B. 0C. 1D. 2


0 0 0 0 0 10 0 0 0 1 00 0 0 1 0 00 0 1 0 0 00 1 0 0 0 01 0 0 0 0 0

is:


A. −1B. 0C. 1D. 2


1 1 42 2 53 3 6

is:

A. −1B. 0C. 1D. 2

For questions 5, 6, 7 and 8: Let P2 be the vector space of polynomialsof degree 2 or less, so vectors have the form f = a0 + a1t+ a2t

2. Considerthe inner product

〈f, g〉 :=

∫ 1

0

f(t)g(t)dt.

For example 〈1, t〉 =∫ 1

01 · t dt =

(12 t

2)|10 = 1

2 .

(53) If f(t) = t, then the length (or norm) ‖f‖ is

A. 1/2

B. 1/√

2

C. 1/3

D. 1/√

3

(54) Let f(t) = t and g(t) = t2 − 34 t. What is the inner product 〈f, g〉?

A. 1/2

B. −3/4

C. 0

D. 1/√

2

(55) If V = Span(f) = Span(t), and g(t) = t2 − 34 t, then the projection g of g

onto V is

A. t2

B. − 34 t

C. 0

D. t2 − 34 t

(56) Still letting V = Span(f) = Span(t) and g(t) = t2 − 34 t, the projection g⊥

of g onto V ⊥ is


A. t2

B. − 34 t

C. 0

D. t2 − 34 t


(57) Which of the following vectors is an eigenvector for the matrix[2 10 1

2

]with eigenvalue 1

2?

A.

[10

]B.

[01

]C.

[− 2

31

]D.

[123

](58) What is the determinant of2 2 4

6 5 00 1 3

?

A. 18

B. −8

C. 22

D. −4

E. 30

(59) Is 1 an eigenvalue of the matrix3 2 46 6 00 1 4

?

(Hint: Compare to the previous problem.)

A. YesB. No


(60) Suppose A is a matrix with real entries, such as

[0 −11 0

]. Then the eigen-

values of A must be real.

A. TrueB. False

(61) Let A be a 3×3 matrix so that A

11−1

=

000

. Then A must have non-zero

determinant.

A. TrueB. False

(62) Consider the matrix

A =

1 1 11 1 −12 −1 0

Then A−1 is equal to AT .

A. TrueB. False

(63) Suppose A is a 2×2 matrix and it has a basis of eigenvectors

[11

]and

[−11

].

Then A must be symmetric.

A. TrueB. False

(64) Suppose A is a 2× 2 matrix and it has a basis of eigenvectors

[11

]and

[10

].

Then A must be symmetric.

A. TrueB. False

(65) Suppose A is a 3× 3 matrix with eigenvalues 0, 2 and 7. Then

(a) A must be invertible(b) A must be non-invertible(c) Not enough information to tell


(66) Suppose A is a 3× 3 matrix so that

142

is an eigenvector with eigenvalue

3 and

010

is an eigenvector with eigenvalue 2. What is A

153

?

(a)

152

(b)

3146

(c) Not enough information to tell.

(67) Which matrix has exactly two eigenspaces: Span

2

10

corresponding to

λ = 3 and Span

0

01

corresponding to λ = 2?

(a)

3 0 00 3 00 0 3

(b)

5 −4 01 1 00 0 2

(c)

5 −4 01 1 00 0 4

(d)

3 0 00 3 00 0 2

(e) None of the above.

(68) Which of the following statements are true?A. If M is a 4 × 4 matrix with eigenvalues 1, 2, 3 and 4, then M must

have an eigenbasis.B. If M is a 4× 4 symmetric matrix with eigenvalues 1, 2, 3 and 3, then

M must have an eigenbasis.

(a) Both A and B are true(b) A is true but B is false(c) B is true but A is false(d) Neither A nor B is true

(69) Suppose A is a 5 × 5 symmetric matrix, and 1 is an eigenvalue with

eigenspace Span

30120

. Suppose the only other eigenvalue of A is 2.

What are the possible dimensions of the eigenspace of 2?

(a) 0 only


(b) 1 only(c) 0, 1, 2, 3, or 4 only(d) 1, 2, 3 or 4 only(e) 4 only

(70) Suppose A is a 2× 2 matrix,

A =

[a bc d

]and that v =

[23

]is an eigenvector with eigenvalue 2. Is v also an eigen-

vector of the matrix 2A, and if so, what is its eigenvalue?

(a) v is not necessarily an eigenvector of 2A.(b) v must be an eigenvector of 2A, with eigenvalue 1/2(c) v must be an eigenvector of 2A, with eigenvalue 2(d) v must be an eigenvector of 2A, with eigenvalue 4

(71) Suppose Q is an m × n matrix with orthonormal columns, and m > n.Which of the following statements must be true?

A. QQT = Im, where Im is the m×m identity matrix.B. QTQ = In, where In is the n× n identity matrix.

(a) Both A and B are true(b) A is true but B is false(c) B is true but A is false(d) Both A and B are false

If f and g are functions defined on [0, 4π] we define their dot product to be

〈f, g〉 =

∫ 4π

0

f(x)g(x) dx.

For numbers 13 and 14 below, consider the functions

f(x) =

{1 x ∈ [0, π)0 x ∈ [π, 4π]

, g(x) = sinx.

(72) What is the dot product 〈f, g〉?

(a) 2π(b) π(c) 2(d) 1(e) 0

(73) Note that 〈g, g〉 = 2π. What is the sinx term of the Fourier series for f?In other words, the orthogonal projection of f onto g?


(a) 2 sinx(b) 1

π sinx

(c) 1√2π

sinx

(d)√

2π sinx

(e) 0

(74) The determinant of the matrix A =

1 0 0 00 1 0 0−3 0 1 00 0 0 1

is:

(a) 3(b) -3(c) 1(d) -1(e) 0

(75) If A is a square matrix and det(A) = 5, then det(2A) must be

(a) 5(b) 10(c) 25(d) 40(e) Not enough information to tell.

(76) If A is a 3× 3 matrix and A

20−1

=

000

then det(A) must be

(a) 0(b) 1(c) -1(d) 2(e) Not enough information to tell.

(77) The eigenvalues of the matrix A =

[1 23 6

]are:

(a) 1 and 0(b) 1 and 6(c) 2 and 3(d) 3.5 and -3.5(e) 0 and 7


(78) If f : R2 → R is a twice-differentiable function, with a critical point at 0,

and its Hessian matrix at this point is H =

[2 −1−1 3

]then

(a) f must have a local minimum at 0(b) f must have a local maximum at 0(c) f cannot have a local minimum or maximum at 0(d) Not enough information to tell

(79) True or False: If A and B are two invertible n× n matrices, then

(AB)−1 = B−1A−1.

(a) True(b) False

(80) True or False: If A, B and C are three n× n matrices, then

(ABC)2 = A2B2C2.

(a) True(b) False

(81) True or False: Every invertible n×n matrix A can be written as a productof elementary matrices:

A = E1E2 · · ·Er(a) True(b) False

(82) True or False: If E is an elementary matrix, then E−1 is also an elementarymatrix.

(a) True(b) False

(83) True or False: If A is an n × n matrix and the columns of A are linearlyindependent, then A is invertible.

(a) True(b) False

(84) True or False: If

[ab

]and

[cd

]are linearly independent, then

[ac

]and

[bd

]must be linearly independent.

(a) True(b) False


(85) True or False: If A is an m×n matrix and an echelon form U has a bottomrow consisting entirely of zeroes, then the columns of A must be linearlydependent.

(a) True(b) False

(86) True or False: If A is an m×n matrix and an echelon form U has a bottomrow consisting entirely of zeroes, then the columns of A do not span all ofRm.

(a) True(b) False

(87) True or False: If V is a vector space of dimension n, and v1, . . . ,vn aren different vectors that together span V , then they must also be linearlyindependent.

(a) True(b) False

(88) True or False: If v and w are perpendicular vectors in R2, then the dotproduct v ·w must be zero.

(a) True(b) False

(89) True or False: If A is an m× n matrix, then the column space Col(A) andthe left null space Nul(AT ) are orthogonal complements.

(a) True(b) False

(90) True or False: If v and w are vectors in Rn, w is in the span of v, and ifw is the projection of w onto v, then

Span{v,w} = Span{v, w}

(a) True(b) False

(91) True or False: If v and w are vectors in Rn, w is not in the span of v, andif w⊥ is the projection of w onto the orthogonal complement of Span{v},then

Span{v,w} = Span{v,w⊥}(a) True


(b) False

(92) True or False: For every square matrix A, both A and AT have the samedeterminant.

(a) True(b) False

(93) True or False: Rearranging the rows of A does not change its determinant.

(a) True(b) False

(94) True or False: If A is a square matrix, then its null space Nul(A) is alsoone of the eigenspaces of A.

(a) True(b) False

(95) True or False: Every square matrix A has a diagonalization A = PDP−1,where D is a diagonal matrix.

(a) True(b) False

(96) True or False: If A is symmetric, then all its eigenvalues must be real.

(a) True(b) False

(97) True or False: If f is a twice-differentiable function with a critical point at0, and every entry in its Hessian matrix H is positive, then f must have alocal minimum at 0.

(a) True(b) False

(98) True or False: If A is a Markov matrix, then A must have λ = 1 as aneigenvalue.

(a) True(b) False


(99) Let

V =

2a+ 3b

2c3a+ c

: a, b, c ∈ R

Which of the following is a basis for V ?

(a)

2

03

,3

00

,0

21

(b) Span

2

03

,3

00

,0

21

(100) The matrix A can be put into Echelon form using the following row oper-

ations:

A =

1 0 3−3 4 −72 4 5

R2→R2+3R1−−−−−−−−−→

1 0 30 4 22 4 5

R3→R3−2R1−−−−−−−−−→

1 0 30 4 20 4 −1

R3→R3−R2−−−−−−−−→

1 0 30 4 20 0 −3

= U

What is the matrix L in the LU decomposition of A corresponding to theabove U?

(a)

1 0 03 1 0−2 −1 1

(b)

1 0 03 1 0−5 −1 1

(c)

1 0 0−3 1 0−1 1 1

(d)

1 0 0−3 1 02 1 1

(101) Suppose A is a 3× 3 matrix with A

132

=

000

. Is A invertible?

(a) Yes(b) No(c) Not enough information to tell

(102) Suppose A is a 3× 3 matrix with A

132

=

132

. Is A invertible?

(a) Yes(b) No(c) Not enough information to tell


(103) Select the inverse of

[3 21 2

]:

(a)

[2 −2−1 3

](b) 1

4

[2 −2−1 3

](c)

[−3 12 −2

](d) 1

4

[−3 12 −2

]

(104) Suppose A is a 3× 3 matrix with columns v1,v2 and v3:

A =

| | |v1 v2 v3

| | |

and Nul(A) = Span

2−31

. Are the columns of A linearly dependent,

and if so, what is a nontrivial dependence relation between them?

(a) The columns of A are linearly independent(b) The columns of A are linearly dependent and a dependence relation is

0v1 + 0v2 + 0v3 = 0

(c) The columns of A are linearly dependent and a dependence relation is

4v1 − 6v2 + 2v3 = 0

(d) The columns of A are linearly dependent, but neither of the aboveoptions is a nontrivial dependence relation

(105) Which of the following are subspaces of the indicated vector space:

A. If A is a 2× 2 matrix, {x | Ax = 3x}, as a subset of R2

B. If A is a 2× 2 matrix, {x | Ax =

[20

]}, as a subset of R2

(a) Only A is a subspace(b) Only B is a subspace(c) Both A and B are subspaces(d) Neither A nor B are subspaces

(106) The matrix

A =

98 −225 −309 1356 −114 −162 22−14 21 33 −1314 −27 −39 7

has Echelon form

U =

7 1 −5 180 1 1 10 0 0 00 0 0 0

.


Which of the following is a basis for the column space of A?

A.

9856−1414

,−225−114

21−27

B.

7000

,

1100

C.

9856−1414

,−225−114

21−27

,−309−162

33−39

,

1322−13

7

(a) A only(b) B only(c) A and B only(d) A and C only

(107) If A is a 3× 6 matrix of rank 2, then Nul(A) has dimension

(a) 0(b) 1(c) 2(d) 3(e) 4

(108) The matrix

A =

2 3 3 44 6 6 82 5 7 7

has Echelon form

U =

2 3 3 40 2 4 30 0 0 0

.Which of the following are a basis for Col(AT )?

A.

2334

,

4668

B.

2334

,

0243


C.

2334

,

4668

,

2577

(a) A only(b) B only(c) A and B only(d) A and C only

(109) For every 5× 5 matrix A, if

dim Col(AT ) = 3,

then the multiplicity of the eigenvalue λ = 0 of A

(a) must be 2.(b) must be 3.(c) can be either 0, 1, or 2.(d) can be either 3, 4, or 5.(e) can be either 2, 3, 4, or 5.

(110) Which of the following maps T : R2 → R2 are linear?

A. T

([xy

])=

[x+ 1x+ 1

]B. T

([xy

])=

[2x− 3y

x

]C. T

([xy

])=

[xy2

]D. Rotation by an angle of α about the origin

(a) A, B, C, and D(b) A, B, and D only(c) B and D only(d) A and B only(e) None of the above

(111) Suppose f : P1 → P1 is a linear map that has matrix

A =

[2 −11 0

]with respect to input and output bases {1, t}. What is f(1 + 3t)?

(a)

[53

](b)

[−11

](c) 3 + 5t


(d) −1 + t

(112) Let L be the linear transformation from P2 to P2 given by

L(p(t)) = p′(t) + 4p(t)

and let B = {1, t, t2} be the standard basis for P2. Then the coordinatematrix LBB representing L with respect to input and output basis B is:

(a)

1 0 04 1 00 8 1

(b)

4 0 01 4 00 2 4

(c)

4 1 00 4 20 0 4

(d)

1 4 00 1 80 0 1

(e) None of the above.

(113) Given v =

1224

and w =

1−111

what is the projection of v onto w?

(a) 54

1−111

(b) 14

−113311

(c) 52

1−111

(d) 15

1224

(e) None of the above

(114) If B = {v1,v2,v3,v4} is an orthogonal basis of R4 and W = Span{v1,v2},then the coordinate matrix PBB representing the projection map onto Wwith input and output basis B is:

(a)

1 0 0 00 0 0 00 0 0 00 0 0 0

(b)

1 0 0 00 1 0 00 0 0 00 0 0 0

(c)

1 0 0 00 1 0 00 0 1 00 0 0 1

(d) None of the above

(115) Consider the orthonormal basis B =

1√2

1−10

, 1√3

111

, 1√6

11−2

of

R3. What are the coordinates of the vector

100

in this basis?

(a)

111

(b) 1√2

1−10

(c)

1/21/31/6

(d)

1√21√31√6

(116) If V is a 13 dimensional subspace of R20, then the dimension of V ⊥ mustbe


(a) 0(b) 7(c) 13(d) 20

(117) If A is the edge-node incidence matrix for the graph

1

2 3

4

>

>

>>>1

2

34

5

then the dimension of Nul(A) is

(a) 0(b) 1(c) 2(d) 4(e) 5

(118) Consider the three row vectors aT ,bT , cT , where a,b, c ∈ R3. Let

A =

aT

bT

cT

be a matrix with determinant 3. What is the determinant of the matrix

B =

2aT

2aT + 2cT

2bT + 10cT

?

(a) -6(b) 6(c) -24(d) 24

(119) IfA and B are bases, and IBA =

[2 31 4

], then the equation

[55

]=

[2 31 4

] [11

]means:

(a) If v has A-coordinates

[11

], then it has B-coordinates

[55

].

(b) If v has B-coordinates

[11

], then it has A-coordinates

[55

].

(c) If v =

[11

], then it has B-coordinates

[55

].


(d) If v =

[11

], then it has A-coordinates

[55

].

linear algebra question bank - binghamton university

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