Deck 4: Subspaces

Question 1

Determine if S is a subspace of R², where S is the subset consisting of all vectors
where

.

Answer 1

S is not a subspace.

Question 2

Determine if S is a subspace of R³, where S is the subset consisting of all vectors
where

.

Answer 2

S is a subspace.

Question 3

Determine if S is a subspace of R³, where S is the subset consisting of all vectors
where

.

Answer 3

S is not a subspace.

Question 4

Determine if S is a subspace of R, where S is the subset consisting of all vectors
where q is a rational number.

Question 5

Determine the null space of

Question 6

Determine the null space of

Question 7

Determine the null space of

Question 8

Determine the null space of

Question 9

Let
for the matrix A. Determine if the vector b is in the kernel of T and if the vector c is in the range of T.

Question 10

Let
for the matrix A. Determine if the vector b is in the kernel of T and if the vector c is in the range of T.

Question 11

If

,

, and
are subspaces of Rⁿ, then their intersection
is also a subspace of Rⁿ.

Question 12

Let A be an
invertible matrix, let b be an
column vector. Let S be the set of all vectors x such that

. Then S is a subspace of Rⁿ.

Question 13

The null space of an
matrix A is a subspace of Rⁿ if and only if A is invertible.

Question 14

Let A be an
matrix, and B an
matrix. Then the null space of B is a subspace of the null space of AB.

Question 15

Let A be an m×nmatrix, and B an m×rmatrix. Then the range of AB is a subspace of the range of

Question 16

By forming matrix rows, find a basis for the given subspace S and give the dimension of S, where

Question 17

By forming matrix rows, find a basis for the given subspace S and give the dimension of S, where

Question 18

By forming matrix rows, find a basis for the given subspace S and give the dimension of S, where

Question 19

Find a basis for the given subspace S by deleting linearly dependent vectors, and give the dimension of S. No actual computation is needed.

Question 20

By forming matrix columns, find a basis for the given subspace S and give the dimension of S, where

Question 21

By forming matrix columns, find a basis for the given subspace S and give the dimension of S, where

Question 22

Expand the given set to form a basis for R³.

Question 23

Expand the given set to form a basis for R⁴.

Question 24

Find a basis for the null space of the given matrix A and give

.

Question 25

Find a basis for the null space of the given matrix A and give

.

Question 26

If
and
are subspaces of Rⁿ, with

, then
is a subset of

.

Question 27

If and
, then .

Question 28

If
and

, and

, then

.

Question 29

If E is an
elementary matrix and A is an
matrix, then the subspace spanned by the columns of A is the same as the subspace spanned by the columns of EA.

Question 30

If E is an
elementary matrix and A is an
matrix, then the subspace spanned by the rows of A is the same as the subspace spanned by the rows of EA.

Question 31

Find bases for the column space of A, the row space of A, and the null space of A.

Question 32

Find bases for the column space of A, the row space of A, and the null space of A.

Question 33

Find bases for the column space of A, the row space of A, and the null space of A.

Question 34

Find all values of
so that rank

, where

Question 35

Find all values of
so that rank

, where

Question 36

Suppose that
is a
matrix. If the dimension of
is

, what are the dimensions of
and

?

Question 37

Suppose that A is a
matrix. If the dimension of
is

, and the dimension of
is

, what is

?

Question 38

Suppose that A is a
matrix and that

. If

, what is the dimension of the kernel of T ?

Question 39

Suppose that A is a
matrix, and that B is an equivalent matrix in echelon form. If B has
pivot columns, what is

?

Question 40

Suppose that A is an
matrix. If

, and

, what is

?

Question 41

If A and B are equivalent matrices, then

.

Question 42

If
and

, then
is a
matrix.

Question 43

If T is a one-to-one linear transformation from R³ to R⁵, and A is a matrix such that

, then

.

Question 44

If T is an onto linear transformation from R³ to R⁵, and A is a matrix such that

, then

.

Question 45

If A is an
matrix such that

, then

.

Question 46

Convert the coordinate vector
from the given basis B to the standard basis.

Question 47

Convert the coordinate vector
from the given basis B to the standard basis.

Question 48

Find the change of basis matrix from the standard basis to B, and then convert x to the coordinate vector with respect to B.

Question 49

Find the change of basis matrix from the standard basis to B, and then convert x to the coordinate vector with respect to B.

Question 50

Find the change of basis matrix from B₁ to B₂.

Question 51

Find the change of basis matrix from B₁ to B₂.

Question 52

Find the change of basis matrix from B₂ to B₁.

Question 53

Find the change of basis matrix from B₂ to B₁.

Question 54

Find the change of basis matrix from B₁ to B₂.

Question 55

Find
given

, where

Question 56

If B is a basis, then
for all scalars

,
and all vectors

,
in span (B).

Question 57

If B₁, B₂, and B₃ are all bases for a given subspace, A is the change of basis matrix from B₁ to B₂, and B is the change of basis matrix from B₂ to B₃, then the change of basis matrix from B₁ to B₃ is given by BA.

Question 58

If B₁, B₂ are bases for a given subspace, and A is the change of basis matrix from B₁ to B₂, then A is invertible, and
is the change of basis matrix from B₂ to B₁.

Question 59

If B is a basis for a given subspace, and u and v are vectors in span (B) such that

, then

.

Question 60

If B is a basis for a given subspace, and u is in span (B) with

, then

.