True / False

Question 1

If a system of linear equations has no free variables, then it has a unique solution.

Answer 1

False

Question 2

If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.

Answer 2

False

Question 3

If A is m × n with m pivots, then the linear transformation x → Ax is one to one

Answer 3

False

Question 4

Let S = {b1 … b2} be linearly independent set in Rn, then S is a basis for Rn

Answer 4

True

Question 5

If A is a 2x2 matrix such that A^2x = 0, then S is a basis for Rn

Answer 5

False

Question 6

If A is a symmetric matrix then A is invertible.

Answer 6

False

Question 7

A system of 4 equations in 5 unknowns can never have a unique solution.

Answer 7

True

Question 8

The null space of an invertible matrix contains only the zero vector

Answer 8

True

Question 9

Two subspaces that meet only in the zero vector are orthogonal.

Answer 9

False

Question 10

If two matrices have the same determinant, they must be similar.

Answer 10

False

Question 11

If a matrix is diagonalizable, it must have distinct eigenvalues

Answer 11

False

Question 12

Suppose we apply an elemantary row operation to a matrix A and obtain matrix B. Then A can be obtained by performing an elemantary row operation on B.

Answer 12

True

Question 13

Elementart row operations on an augmented matrix changes the solution set of the linear system

Answer 13

False

Question 14

A consistant system of linear equations has on or more solution

Answer 14

True

Question 15

Suppose a 3 x 5 coefficient matrix for a system has three pivot columns. Is the system consistent?

Answer 15

Yes

Question 16

Suppose a system of linear equations has a 3 x 5 augmented matrix whose fifth column pivot columns. Is the system consistent?

Answer 16

No

Question 17

The echelon form of a matrix is unique

Answer 17

False

Question 18

The vector u resulsts when a vector u - v is added to the vector V

Answer 18

True

Question 19

An example of a linear combination of vectors v1 & v2 is the vector (1/2)v1

Answer 19

True

Question 20

Every matrix equation Ax = b corresponds to a vector equation with the same solution set

Answer 20

True

Question 21

If A is a 3x3 matrix then det (5A) = 5 det (A)

Answer 21

False

Question 22

if u a v are in R2 and det[uv] = 10, the area of the triangle in the plane with vertices at 0, u, v is 10

Answer 22

False

Question 23

if A^3 = 0 then det(A) = 0

Answer 23

True

Question 24

if S is a linearly dependent set, then each vector is a linear combination of the other vectors in S

Answer 24

False

Question 25

The columns of any 4 x 5 matrix are linearly dependent

Answer 25

True

Question 26

if v1, v2, v3 and v4 are linearly independent vectors in R4, then {v1, v2, v3} is also linearly independent.
Hint: think about x1v1 + x2v2 + x3v3 + x4v4 = 0

Answer 26

True

Question 27

The codomain of the transformation x –> Ax is the set of all linear combinations of the columns of A

Answer 27

False

Question 28

If A is a m x n matrix, then the domain of the transformation x –> Ax is Rn

Answer 28

True

Question 29

If A is a m x n matrix, then the range of the transformation x –> Ax is Rm

Answer 29

False

Question 30

If A is a 3x2 matrix, then the transformation x –> Ax with co-domain R3 cannot be onto

Answer 30

True

Question 31

If A is a 2x3 matrix, then the transformation x –> Ax can’t be one-t-one

Answer 31

True

Question 32

If A is a 3x2 matrix, then the transformation x –> Ax can’t be one-t-one

Answer 32

False

Question 33

the canonical vectors of Rn form and orthonormal basis of Rn

Answer 33

True

Question 34

u, v & w are non zero vectors
If u & v are both orthogonal to w, then u & v are aligned (there exists c such as u =cv)

Answer 34

True

Question 35

For any nxn matrix A& vectors u, v in Rn we have (Au).(Av) = u.v

Answer 35

False

Question 36

(1,1,0,1) is a unit vector in the same direction as (3,3,0,3)

Answer 36

False

Question 37

Not every linearly independent set in Rn is an orthogonal set

Answer 37

True

Question 38

The first row of AB is the first row of A multiplied on the right by B

Answer 38

True

Question 39

A square matrix with two identical columns can be invertible

Answer 39

False

Question 40

If A is a square matrix such that the equation Ax = b is consistent for b in Rn, then A is invertible

Answer 40

True

Question 41

The matrix {(1,0,0)(0,1,0)(0,a,1)} is invertible regardless the value of a

Answer 41

True

Question 42

For any matrix A, the products AA^T & A^TA are well defined

Answer 42

True

Question 43

if A is an nxn matrix, then (A^2)^T = (A^T)^2

Answer 43

True

Question 44

A vector is an arrow in 3D space

Answer 44

False

Question 45

A subset H of a vector space V is a subspace of V if the zero vector is in H

Answer 45

False

Question 46

R2 is a subspace of R3

Answer 46

False

Question 47

If there is a set {v1,…,v2} that spans V, then dim(V) <= r

Answer 47

True

Question 48

if dim V = p, then there exists a spanning set of p+1 vectors in V

Answer 48

True

Question 49

if p >= 2 & dim V = p, then every set of p - 1 non zero vectors is linearly independent

Answer 49

False