Question 1

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

Accepted Answer

A) True 
 B)False  True

Question 2

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

Accepted Answer

Column space basis: 
   
 ; row space basis: 
   
 ; null space basis:

Question 3

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

Accepted Answer

S is not a subspace.

Question 4

Expand the given set to form a basis for R3.

Accepted Answer

Question 5

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

Accepted Answer

The answer of Suppose that A is a 
 ...

Question 6

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 not in the kern

Question 7

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 8

Suppose that A is a 
   matrix and that 
   
. If 
   
, what is the dimension of the kernel of T ?

Accepted Answer

The answer of Suppose that A is a 
 ...

Question 9

Suppose that 
   is a 
   matrix. If the dimension of 
   is 
   
, what are the dimensions of 
   and 
   
?

Accepted Answer

The answer of Suppose that 
   is a...

Question 10

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.

Accepted Answer

A) True 
 B)False

Question 11

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.

Accepted Answer

A) True 
 B)False

Question 12

Find the change of basis matrix from B2 to B1.

Accepted Answer

Question 13

Suppose that A is a 
   matrix. If the dimension of 
   is 
   
, and the dimension of 
   is 
   
, what is 
   
?

Accepted Answer

The answer of Suppose that A is a 
 ...

Question 14

Find the change of basis matrix from B1 to B2.

Accepted Answer

Question 15

If A and B are equivalent matrices, then 
   
.

Accepted Answer

A) True 
 B)False

Question 16

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

Accepted Answer

The answer of Find a basis for the null space...

Question 17

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

Accepted Answer

Basis: 
 @#IMG-DLM&

Question 18

If A is an 
   matrix such that 
   
, then 
   
.

Accepted Answer

A) True 
 B)False

Question 19

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

Accepted Answer

The answer of Convert the coordinate vector 
  ...

Question 20

If 
   and 
   are subspaces of Rn, with 
   
, then 
   is a subset of 
   
.

Accepted Answer

A) True 
 B)False

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

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

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

Expand the given set to form a basis for R³.

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

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

Suppose that A is a matrix and that . If , what is the dimension of the kernel of T ?

Suppose that is a matrix. If the dimension of is , what are the dimensions of and ?

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.

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.

Find the change of basis matrix from B₂ to B₁.

Suppose that A is a matrix. If the dimension of is , and the dimension of is , what is ?

Find the change of basis matrix from B₁ to B₂.

If A and B are equivalent matrices, then .

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

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

If A is an matrix such that , then .

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

If and are subspaces of Rⁿ, with , then is a subset of .

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

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

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

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

Expand the given set to form a basis for R3.

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

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

Suppose that A is a matrix and that . If , what is the dimension of the kernel of T ?

Suppose that is a matrix. If the dimension of is , what are the dimensions of and ?

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.

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.

Find the change of basis matrix from B2 to B1.

Suppose that A is a matrix. If the dimension of is , and the dimension of is , what is ?

Find the change of basis matrix from B1 to B2.

If A and B are equivalent matrices, then .

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

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

If A is an matrix such that , then .

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

If and are subspaces of Rn, with , then is a subset of .

Systems of Linear Equations

Euclidean Space

Matrices

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

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

Expand the given set to form a basis for R³.

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.

Find the change of basis matrix from B₂ to B₁.

Find the change of basis matrix from B₁ to B₂.

If and are subspaces of Rⁿ, with , then is a subset of .