Question 1

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.

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

The answer of Find the change of basis matrix from...

Question 3

Find the change of basis matrix from B2 to B1.

Accepted Answer

Question 4

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.

Accepted Answer

A) True 
 B)False

Question 5

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.

Accepted Answer

b is in the kernel o

Question 6

Find all values of 
   so that rank 
   
, where

Accepted Answer

The answer of Find all values of 
  ...

Question 7

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

Accepted Answer

Basis: 
 @#IMG-DLM&

Question 8

If 
   and 
   
, and 
   
, then 
   
.

Accepted Answer

A) True 
 B)False

Question 9

If   and   
, then   .

Accepted Answer

A) True 
 B)False

Question 10

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

Accepted Answer

Basis: 
 @#IMG-DLM&

Question 11

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

Accepted Answer

Basis: 
 @#IMG-DLM&

Question 12

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

Accepted Answer

S is not a

Question 13

Find the change of basis matrix from B1 to B2.

Accepted Answer

Question 14

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

Accepted Answer

S is not a

Question 15

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

Accepted Answer

Column space basis:

Question 16

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

Accepted Answer

A) True 
 B)False

Question 17

Suppose that A is an 
   matrix. If 
   
, and 
   
, what is 
   
?

Accepted Answer

The answer of Suppose that A is an 
 ...

Question 18

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

Accepted Answer

A) True 
 B)False

Question 19

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.

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

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.

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

Find the change of basis matrix from B₂ to B₁.

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.

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.

Find all values of so that rank , where

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

If and , and , then .

If and , then .

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

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

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

Find the change of basis matrix from B₁ to B₂.

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

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

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

Suppose that A is an matrix. If , and , what is ?

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

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.

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

Systems of Linear Equations

Euclidean Space

Matrices

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

Exam 4: Subspaces

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.

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

Find the change of basis matrix from B2 to B1.

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.

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.

Find all values of so that rank , where

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

If and , and , then .

If and , then .

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

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

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

Find the change of basis matrix from B1 to B2.

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

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

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

Suppose that A is an matrix. If , and , what is ?

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

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.

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

Systems of Linear Equations

Euclidean Space

Matrices

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

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.

Find the change of basis matrix from B₂ to B₁.

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.

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

Find the change of basis matrix from B₁ to B₂.

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

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

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