Test 2

Question 1

If f is a function in the vector space V of all real-valued functions of R and if f(t)=0 for some t, then f is the zero vector in V.

Answer 1

FALSE. while f(t) = 0 for some t might be a property of a function, it doesn’t guarantee that the function is the zero vector in the vector space

Question 2

The Null Space of A is the solution set of the equation Ax = 0

Answer 2

TRUE

Question 3

Col A is the set of all SOLUTIONS to Ax = b

Answer 3

FALSE - Col(A) determine all the possible OUTPUT, not solution

Question 4

If the equation Ax = b is consistent, then Col(A) = R^m

Answer 4

FALSE - Col(A) might not Span all of R^m.

Question 5

The RANGE of a linear transformation is a vector space

Answer 5

TRUE - also the range is a subspace of the output vector space W.

Question 6

A single vector by itself is linear dependent

Answer 6

FALSE - single vector by itself cannot be expressed as a linear combination of any other vectors, including itself (with a zero scalar).

Question 7

If H = Span{b1…bp}, then {b1…bp} is a basis for H

Answer 7

TRUE

Question 8

If V = R^n and C is the std basis for V, then P(C<-B) is the same as the change of coordinates matrix PB.

Answer 8

TRUE

Question 9

If v1…vp are in R^n , then Span{v1…vp} is the same as the column space of the matrix [v1…vp]

Answer 9

TRUE

Question 10

If B ={v1…vp} is a basis for a subspace H and if x =c1v1 +…+cpvp, then c1,…cp are the coordinates of x relative to the basis B.

Answer 10

TRUE

Question 11

The dimension of Col A is the number of pivot columns of A.

Answer 11

TRUE

Question 11

Each line in R^n is a one-dimensional subspace of R^n

Answer 11

FALSE - could be 0

Question 12

If P[B] is the change-of-coordinates matrix, then [x]B = P[B].x, for x in V

Answer 12

FALSE - b/c [x]B = x . (P[B])^-1

Question 13

A plane in R^3 is a two-dimensional subspace of R^3

Answer 13

TRUE

Question 14

The dimension of the vector space of signals S is 10

Answer 14

FALSE

Question 15

Given A an n x n invertible matrix, the following are equivalent

Answer 15

1. Columns of A form a basis of R^n
2. Col A = R^n
3. Rank A = n
4. Nullity A = 0
5. Nul(A) = {0}

Question 16

If V = R^n and C is the std basis for V, then P(C<-B) is the same as the change of coordinates matrix P(B) in 4.4

Answer 16

TRUE

Question 17

An n x n determinant is defined by determinants of (n-1) x (n-1) submatrices.

Answer 17

FALSE - missing cofactor

Question 18

a row replacement operation does not affect the determinant of a matrix

Answer 18

TRUE - row replacement det A = det B

Question 19

If the columns of A are linearly dependent , then det A = 0

Answer 19

TRUE - colinear lines will have zero area (det = 0)

Question 20

If three row interchanges are made in succession, then the new determinant equals the old determinant

Answer 20

FALSE - each row interchanges multiply det by (-1); three times will made det A = - det B

Question 21

The set of 2x2 matrices with positive determinant is a subspace

Answer 21

FALSE - not enough information to satisfy conditions:
1. Closure under addition det(A+B) > 0
2. A + B > 0

Question 22

The set of 2x2 matrices with zero determinant is a subspace

Answer 22

FALSE - det(A+B) might not = 0 and also linear dependent row/col might not sustain once added

Question 23

If A is a 2x2 matrix with a zero determinant, then one column of A is a multiple of the other

Answer 23

TRUE - det = 0 means linear dependent row/col

Question 24

If B is produced by multiplying row 3 of A by 5, then det B = 5. det A

Answer 24

TRUE

Question 25

det ( A^T . A ) >= 0

Answer 25

TRUE when A is positive definite matrices

Question 26

The set of all linear combinations of v1,…,vp is a vector space

Answer 26

TRUE

Question 27

If S is linearly independent, then S is a basis for V

Answer 27

FALSE - only one condition satisfied, S also need to span space V

Question 28

If Span S = V, then some subset of S is a basis for V

Answer 28

TRUE

Question 29

Determine if w = [. . .] is in subspace of R^3 created by v1=[. . .] and v2=[. . .]

Answer 29

w must be in Span (v1,v2) or w must be the result of a linear combination of v1,v2 ( c1v1 + c2v2 = w)

Question 30

Determine given sets { [. .]; [. .] } that can become basis for R^2 or R^3

Answer 30

1. Check det; if det is non-zero -> invertible -> sets are basis for R^2 / R^3
2. Must have n pivot position in n row for R^n

Question 31

Rank of matrix A is dimension of pivot column space A

Answer 31

TRUE - n pivot columns -> rank = n

Question 32

Rank(A) + dim Nul(A) = n

Answer 32

TRUE - n = total number of columns in matrix A

Question 33

Find coordinate of x in subspace B generated by basis b1,b2

Answer 33

by finding scalars c1.b1 + c2.b2 = [x1; x2]
[x]B = [x1; x2]