Linear Algebra (for Engineering)

Instructions:

This exam is meant to enhance your understanding of the subject matter, and contains roughly the same amount of multiple choice, fill-in-the-blank and true or false questions. For each question, select or fill in the choice that is most appropriate.  

Exam-takers should read through each question carefully and take notes on difficult concepts.


Notation notes:


- S
ome matrices expressed explicitly may have rows separated by semi-colons (;). For example, [1 2 3; 4 5 6] is a 2 x 3 matrix with 1, 2, 3 in the first row and 4, 5, 6 in the second row.

- Unless stated otherwise, matrices given in the problem statement by only representative form (i.e. matrix A along with the size are the only information given versus the the actual components of the matrix) are nonsymmetrical, not diagonal, non-stochastic, non-regular, and are not an identity matrix and do not have any other known matrix numerical patterns. 

-
The matrix I will always be the identity matrix, and the size will be given if necessary to solve the problem. 

- Vectors are represented by bold lower-case letters. For example x, b and are all vectors, if it is a representative set or list, it will be italicized, like x

- Matrices are represented by italicized upper-case letters. For example, A, B and I  are matrices, bold-face will be used sometimes to decrease visual confusion with vectors and other terms. 

- Unless otherwise stated, for multiple choice there is only 1 correct solution.

 

1.

3x + 7y = 11y = 3 + xis an example of a  system is a system of equations with at least one solution.

2.

For the following linear system of equations, which is the echelon form of the  augmented matrix for the system?

 
 

a)
[ 1 3; 7 9 ]
b)
[ 1 3 5; 0 1 2 ]
c)
[ 1 3 5; 7 9 11 ]
d)
[ 1 3 5; 0 1 11 ]

3.

From the choices, which of the following groups of equations contain all of the linear equations below?

I.  

II.  

III.

IV.

V.  

a)
I, II, III, IV and V
b)
II only
c)
II, III and IV
d)
II, IV and V

4.

What requirements must be met in order for a matrix to be in echelon form?

There may be more than one answer, select all that apply.

 

a)
The leading number in each row must be 1
b)
The final number in each row must be 0
c)
All entries directly below a leading number must have only 0
d)
Leading entries are to the right of leading entries in any row above.

5.

Matrices can be row equivalent to more than one reduced echelon matrix.   

a)
True
b)
False

6.

In a solution of a system of linear equations, variables that do not have restrictions imposed on them are specified as variables.

7.

Vectors in R² can be geometrically added by using what shape in a coordinate plane?

 

a)
Rectangle
b)
Parallelogram
c)
Circle
d)
Triangle

8.

A vector equation    has the same solution set as the linear system whose augmented matrix is which of the following? Where and b are vectors composed of real numbers.

a)
[x_1 x_2 ... x_n b]
b)
[x_1 x_2 ... x_n ]
c)
[a_1 a_2 ... a_n b]
d)
[a_1 a_2 ... a_n ]

9.

If   are in Rn , then the set of all linear combinations of   is denoted by { }.  

10.

A set of 3 vectors always spans R

a)
True
b)
False

11.

Which of the following is an identity matrix?  

a)
[1 0 0 ; 0 0 0]
b)
[1 0 0 ; 0 1 0]
c)
[0 0 1 ; 0 1 0 ; 1 0 0]
d)
[1 0 0 ; 1 0 0 ; 1 0 0]
e)
[1 0 0 ; 0 1 0 ; 0 0 1]

12.

Consider the matrix equation Ax = b, where A is an m x n matrix, with columns.  If x is a vector in , then, Ax is defined only if the number of rows of A equals the number of entries in x.

a)
True
b)
False

13.

What is the correct form to solve for the vector x (in in the matrix equation Ax = b, where A is an invertible m x n matrix and b is a vector in ?  

a)
x=A/b
b)
x=A^(-1) b
c)
x= bA^(-1)
d)
All of the above forms
e)
None of the above forms

14.

A is a solution where all of the variables are equal to zero.  

15.

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

a)
True
b)
False

16.

A list of vectors  in Rn  are said to be if the vector equation  has only the trivial solution.

17.

An indexed set  of two or more vectors is linearly dependent if and only if all vectors in  can be written as linear combinations of the others.  

a)
True
b)
False

18.

A or mapping of T from Rn to Rm is a rule that assigns to each vector x in Rn  a vector T(x) in Rm.  

19.

Let 
,    ,   

( ) is the vector in R2 that creates b under an image of T   Note: enter the solution in the following form [0; 0] to represent the vector

20.

The transformation T : R2   R2 defined by T(x) = A(x) is called a transformation.

21.

Let T be the transformation matrix, and u and v vectors in Rn and c a scalar;  ( ) transformations have the property that T(u + v) = T(u) + T(v) and T(cu) = cT(u)

22.

Let A be an m x n matrix and B an n x p matrix: which of the following is the size of the resulting matrix of the product AB ?

a)
m x p
b)
n x n
c)
(m+n) x (n+p)
d)
None of the above

23.

A  matrix is an n x n matrix where all entries except elements aij are zero. Let aij be the element in row i column j of the matrix, with i and j being nonzero. 

24.

Matrix addition can only be performed for matrices that are equal (that have the same number of rows and columns).

a)
True
b)
False

25.

Let A be an m x n  matrix and B an n x p matrix so that AB =  AxB  and BA = BxA.  Does AB = BA?

a)
True
b)
False

26.

Cancellation laws hold for matrix multiplication. That is, if AB = AC, then in general it is true that B = C.   

a)
True
b)
False

27.

Given A, an m x n matrix, the of A is the n x m matrix denoted by AT . The columns of AT are formed from the corresponding rows of A.

28.

If there is a matrix C, and AC = I and CA= I,  then C is the transpose of matrix A.  

a)
True
b)
False

29.

Which of the following is the determinant of   ?

a)
[3 ; 11]
b)
[16 -2]
c)
120
d)
-120

30.

For two matrices, A and B det(AB) = det(A)det(B) = det(BA) is always true.

a)
True
b)
False

31.

If a matrix is invertible, then so is AT.

a)
True
b)
False

32.

Application to computer graphics: What type of movement is represented by the form ?  

a)
Translation
b)
Shear
c)
Shift
d)
Scale

33.

Which of the following matrices represent a counterclockwise rotation about the origin in R2 with angle
(Hint: Draw a coordinate plane and rotate the x and y axis counterclockwise and create column vectors from the ending points.)

a)
[cosâˆ... ; sinâˆ...]
b)
[sinâˆ... ; cosâˆ...]
c)
[cosâˆ... ; -sinâˆ...]
d)
[cosâˆ... -sinâˆ... ; sinâˆ... cosâˆ...]
e)
[cosâˆ... -cosâˆ... ; sinâˆ... -sinâˆ...]

34.

Analogous to the 2D case, 3D homogeneous coordinates take the form (x, y, z, 1) for the point (x, y, z) in R3.  

a)
True
b)
False

35.

In order for a set of vectors, H, to be a subspace of Rn , it must contain the zero vector.  

a)
True
b)
False

36.

The of a matrix A is the set of Col A of all linear combinations of A's columns.  

37.

Unlike a column space, the Null Space may not be a subspace.

a)
True
b)
False

38.

Which of the following forms a basis of Nul A from row reductions if
 

a)
Nul A = Span{ [1 ; 2 ; 0 ; 1 ; 0] , [3 ; 7 ; 1 ; 0 ; 1 ] }
b)
Nul A = Span{ [1 ; -2 ; 0 ; 1 ; 0] , [1 ; 1 ; -1 ; 0 ; 1] }
c)
Nul A = Span{ [2 ; 1 ; 0 ; -1] , [1 ; 1 ; 1 ; 1] , [1 ; 6 ; 4 ; 2] , [4 ; 3 ; 2 ; 1] , [-2 ; 4 ; 3 ; 2] }

39.

The pivot columns of a matrix A form a basis for the of A.

40.

Given the matrix A (and an echelon form of A); which of the following group of vectors forms a complete basis of Col A?    

a)
{ [1 ; 3 ; 2 ; 5] , [3 ; 9 ; 6 ; 15] , [2 ; 1 ; -1 ; 0] , [-6 ; 5 ; 9 ; 14] }
b)
{ [1 ; 3 ; 2 ; 5] , [2 ; 1 ; -1 ; 0] , [-6 ; 5 ; 9 ; 14] }
c)
{ [3 ; 9 ; 6 ; 15] }
d)
{ [1 ; 3 ; 2 ; 5] , [3 ; 9 ; 6 ; 15] , [-6 ; 5 ; 9 ; 14] }

41.

If a vector space V has a basis composed of n vectors, then any set of vectors in V with more than n vectors must be linearly .  

42.

Let ;  w is in Span{ 

a)
True
b)
False

43.

What is always a difference between a basis for a subspace and a spanning set?

a)
The basis is always composed of unit vectors
b)
Vectors in the subspace can be written in only one way as a linear combination of basis vectors
c)
The spanning set may not be able to define all vectors in the subspace
d)
A spanning set always includes the zero vector

44.

The of a nonzero subspace H, is the number of vectors in any basis of H.

45.

The rank of a matrix A, (denoted by rank A) is the dimension of the space of A.  

46.

Which of the following is always the same as rank A?  

a)
dim Nul A
b)
The number of columns in A
c)
The number of rows in A
d)
dim Col A

47.

For an m x n matrix A, Col A exists in the same Euclidean space as the number of of A.

48.

Which of the following is the dimension of the subspace H, which includes the vectors: 

a)
1
b)
2
c)
3
d)
9

49.

A basis for a vector space V is defined as a list of elements of V {} in which they span V, or a spanning set. This means every y in V can be written as: .  If the elements are vectors, they need to be linearly independent to get a basis.  True of false: The coefficients  are known as the coordinates of y in V.

a)
True
b)
False

50.

Which of the following is not a requirement or axiom of a vector space? Let u & v be vectors in the vector space V & c and d real scalars. 

a)
u + v is in V
b)
u + v = v + u
c)
uv is in V
d)
c(u + v) = cu + cv
e)
c(du) = (cd)u

51.

The vector space R2 is a subspace of  R3

a)
True
b)
False

52.

The or null space of a linear transform T is the set of all u in a vector space V , such that T(u) = 0.

53.

Consider two bases  and   for a vector space V such that . Further, suppose With the given information, which of the following vectors represent the coordinate vector of x with respect to the basis C?

a)
[ 6 ; 9 ]
b)
[ 4 ; 1 ]
c)
[ 3 2 ; -6 1 ]
d)
[ 3 -6 ; 2 1 ]

54.

If is the unique n x n matrix that transforms a vector under basis B to a new basis C, or the change-of-coordinates matrix from B to C, then its columns are always linearly independent.  

a)
True
b)
False

55.

If is the change-of-coordinates matrix from B to C, then -1 is the change of coordinates matrix from C to B.  

a)
True
b)
False

56.

If S is a vector space of discrete-time signals, then a in S is a function defined only on the integers and is represented visually as a sequence of numbers.

57.

Given scalars  with  non-zero, and given a signal , the linear difference equation  holds true. True of false: In digital signal processing, the difference equation above is the linear filter with 

a)
True
b)
False

58.

Which of the following present some solutions of the homogeneous difference equation:  

a)
1, -2, 3
b)
1, 2, 3
c)
1^k, (-2)^k, 3^k
d)
1^k, -2^k, 3^k
e)
1^k, 2^k, 3^k

59.

 If  and then  is valid for all k, and has a unique solution whenever  are specified. Let abe scalars and y& zbe functions.

a)
True
b)
False

60.

The signal  satisfies the difference equation   .

a)
True
b)
False

61.

A lightweight cantilever beam is supported at N points spaced 10 ft apart, and has a weight of 250 lb placed 10 ft from the first support.  Let  be the bending moment at the kth support. Then   Suppose the beam is rigidly attached at the Nth support, so the bending moment there is zero. Given that in between, the moments satisfy the three-moment equation , which of the following is the general solution that satisfies the given boundary equations  ?

a)
y_k=c_1 (-1)^k+c_2 (-3)^k
b)
y_k= (-1)^k+(-3)^k
c)
y_(k+2)=c_1 (-1)^k+c_2 (-3)^k
d)
y_(k+1)=c_1 (-1)^k+c_2 (-3)^k
e)
y_(k+2)=(-1)^k+(-3)^k

62.

A matrix is a square matrix whose columns are probability vectors, which are vectors with non-zero entries that add sum to 1.

63.

If P is a stochastic matrix, then what type of vector q for P is represented by the equation   Pq = q?

a)
Stochastic vector
b)
Perfect vector
c)
Equalizing vector
d)
Steady-state vector
e)
Transient vector

64.

Let P = , is the steady-state vector for P. Note: for the vector x =  input [-.2 ; -1] for the solution, there is a space buffering each integer from the vector punctuation marks.

65.

An of an n x n matrix A is a nonzero vector x, such that Ax = x for some scalar  

66.

Let   and   is u an eigenvector of A?

a)
True
b)
False

67.

Which of the following scalars is an eigenvalue of the following matrix A ?  A =

a)
8
b)
7
c)
2
d)
3

68.

The eigenvalues of a matrix are the entries on its main diagonal.

69.

If  are eigenvectors that correspond to distinct eigenvalues λ1, … λr of an n x n matrix A, then the set { is linearly .  

70.

Which of the following choices are the two eigenvalues for the matrix  

a)
3, 15
b)
1, 15
c)
1, 16
d)
15, 16
e)
11, 15

71.

Let A be an n x n matrix and U be any echelon form obtained from A by row replacements and row interchanges (without scaling), and let r be the number of such row interchanges. Then the determinant of A, written as det A is (-1)r times the product of the entries in U.

72.

Let A be an n x n matrix. A row replacement operation on A changes the magnitude of the determinant of A.

a)
True
b)
False

73.

If A and B are n x n matrices, then A is to B if there is an invertible matrix P such that or .  

74.

If D is a diagonal matrix: , then which of the following represents Dk?  

a)
[7^k 0 ; 0 3^k ]
b)
[3^k 3^k ; 7^k 7^k ]
c)
[7^k+3^k 0 ; 0 3^k+7^k )]
d)
[7^k+3^k 3^k ; 7^k 3^k+7^k )]

75.

A square matrix A is said to be if A is similar to a diagonal matrix, that is, of the form  for some invertible matrix P and some diagonal matrix D.  

76.

The following matrix is diagonalizable  

a)
True
b)
False

77.

When A (an nxn matrix) is diagonalizable but has fewer than n distinct eigenvalues, it may still possible to build P in a way that makes P automatically invertible.

a)
True
b)
False

78.

Suppose is a basis for V,  and is a basis for W. Further, let   be a linear transformation with the property that Which of the following is the matrix M for T relative to the bases B and ?

a)
[ 8 -6 5 ; 4 7 -4]
b)
[12 1 1]
c)
[ 8 4 ; -6 7 ; 5 -4 ]
d)
[ 12 ; 1 ; 1 ]

79.

Suppose  where D is a diagonal n x n matrix. If B is the basis for Rn formed from the columns of P, then what matrix is the B-matrix for the transformation

a)
PDP^-1
b)
D
c)
DP^-1
d)
PD
e)
P^-1

80.

For the given eigenvalues below, what is the closest description of the behavior of the curve in a complex-coordinate plane if successive images are taken on a vector x. Let , which are solutions to the differential equation x' = Ax. Hint: the solutions include both real and complex numbers. 

a)
Constant circle
b)
Constant ellipse
c)
Outward spiraling curve
d)
Inward spiraling curve

81.

If x and y are vectors and x·y = 0, then the two vectors are to each other.

82.

A set is an orthonormal set if it is an orthogonal set of vectors.

83.

Let U be an m x n matrix with orthonormal columns, and let x be in Rn. Given this information, is this statement true or false:   =  ?

a)
True
b)
False

84.

Given the basis  for a subspace W of R4

and given that and  

And knowing that  , a third vector will complete an orthogonal basis for W with  and

*Hint: Vector 2 was found using the Gran-Schmidt Process, find a third vector orthogonal to the previous two by completing the Gram-Schmidt prcoess on the third vector of the basis. Also, if the vector is in the form fill in the blank as [ -1 ; -1 ; 1 ; 1 ], with the entries separated from the brackets and semi-colons with spaces, but attached to their negative signs if applicable.

85.

If A is an m x n matrix and b is in Rm, but  is an inconsistent system, a - solution of  is a vector  in Rsuch that:    for all x in Rn

86.

Which of the following is a least squares solution to the unsolvable system    when     has one column,   and   is 1 x 1?

a)
[ 8 ; 10 ]
b)
[ 10 ; 8 ]
c)
4/5
d)
5/4

87.

For the Markov chain   given the matrix
and its eigenvectors   and If it is given that  and, which of the following is the length  ?

a)
2
b)
2^(.5)
c)
-1
d)
0.000014142
e)
2.0*10^(-10)

88.

Which of the following is the number k so that the functions  and  are orthogonal for  and  in C[0,1] ? Hint: Do this by finding the inner product for two functions on the same domain.

a)
-3/4
b)
2
c)
-1
d)
0
e)
1/2

89.

A symmetric matrix is a matrix such that = .

90.

If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.  

a)
True
b)
False

91.

Which of the following is the symmetric matrix of the quadratic form A for which the quadratic function  is equal to  

a)
[ 6 0 0 ; 0 5 0 ; 0 0 7 ]
b)
[ 6 4 -4 ; 4 5 0 ; -4 0 7 ]
c)
[ 6 2 -2 ; 2 5 0 ; -2 0 7 ]
d)
[ 6 2 -2 ; 2 5 0 ; -2 0 7 ]

92.

    is an orthogonal matrix and  is a diagonal matrix. If  ,  do these matrices satisfy the spectral decomposition ?

a)
True
b)
False

93.

Let    Which of the following are the singular values of A? Please note that there may be more than 1 answer, select all of the singular values.

a)
1
b)
2*sqrt(2)
c)
sqrt(3)
d)
4
e)
8

94.

The functions  are linearly independent, and W denotes the   Let .  Is p in W?

a)
True
b)
False

95.

Consider the Markov matrix M and its eigenvectors :     Given:    = ?

Note: this will be a matrix. Enter it in the form [ -x -x ; x  x ] for the matrix 

96.

Given the augmented matrix 1010320101-120000000000 , which of the following are the free variables? Note: There may be more than one answer; select all of the choices that are valid solutions.

a)
x_1
b)
x_2
c)
x_3
d)
x_4

97.

The following matrix has been reduced to reduced echelon form. Which of the following vectors are in the null space M?


Note: new rows in a matrix (or vector) are denoted with a semi-colon (;)

a)
[ 3 ; 4 ; 0 ; 0 ; 1 ]
b)
[ -3 ; 0 ; 1 ]
c)
[ 0 ; 0 ; 0 ; 1 ]
d)
[ 0 ; -8 ; 0 ; 0 ; 2 ]

98.

For an n x n matrix A, if there is exactly one solution b to the equation Ax = b, then A is invertible.  

a)
True
b)
False

99.

Let  be a basis for a vector space V. Then the coordinate mapping  is what type of linear transformation from V onto Rn ?  

a)
one-dimensional
b)
n-to-one
c)
one-to-one
d)
one-to-n

100.

Any basis for a vector space V has the same cardinality as any other basis for that same vector space.

a)
True
b)
False

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