Question 1

Find bases for the four fundamental subspaces of the matrix   .

Accepted Answer

A)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
B)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
C)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
D)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
E)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
A)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
B)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
C)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
D)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   
E)  N(
A) -basis:   , N(AT)-basis:   , R(
A) -basis:   , and R(AT)-basis:   C

Question 2

Find an orthonormal basis for the solution space of the following homogeneous system of linear equations.

Accepted Answer

A)    
B)     
C)     
D)    
E)    
A)    
B)     
C)     
D)    
E)    E

Question 3

Find   ,where    .

Accepted Answer

A)     
B)     
C)     
D)     
E)    
A)     
B)     
C)     
D)     
E)    C

Question 4

S is the subspace of R3 consisting of the xz-plane. Find the orthogonal complement S.

Accepted Answer

A)     
B)     
C)    
D)     
E)     
A)     
B)     
C)    
D)     
E)

Question 5

Which of the following is a description of the orthogonal complement of

Accepted Answer

A)     
B)     
C)     
D)    
E)    
A)     
B)     
C)     
D)    
E)

Question 6

Find the distance d between u=   and v=   .

Accepted Answer

A)     
B)     
C)    
D)    
E)     
A)     
B)     
C)    
D)    
E)

Question 7

Find the least squares regression line for the points   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 8

Find   for the inner product   defined in R3, where     .

Accepted Answer

A)     
B)     
C)     
D)   
E)     
A)     
B)     
C)     
D)   
E)

Question 9

Find the coordinates of   relative to the orthonormal basis   in R3.

Accepted Answer

A)     
B)    
C)    
D)     
E)    
A)     
B)    
C)    
D)     
E)

Question 10

Use the Gram-Schmidt process to change the basis   into an orthonormal basis for the subspace of R3 spanned by the vector   .

Accepted Answer

A)     
B)     
C)     
D)    
E)     
A)     
B)     
C)     
D)    
E)

Question 11

Find the vector v with length 2 and the same direction as the vector   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 12

Use the functions   in C[-1, 1] to find   for the inner product   .

Accepted Answer

A)       
B)       
C)       
D)       
E)       
A)       
B)       
C)       
D)       
E)

Question 13

Find the length of the vector

Accepted Answer

A)    
B)    
C)    
D)    
E)     
A)    
B)    
C)    
D)    
E)

Question 14

Find   for the inner product   defined in R2, where   .

Accepted Answer

A)      
B)       
C)       
D)       
E)       
A)      
B)       
C)       
D)       
E)

Question 15

Use the functions   in C[-1, 1] to find   for the inner product   .

Accepted Answer

A)     
B)     
C)     
D)    
E)     
A)     
B)     
C)     
D)    
E)

Question 16

​Show that the cross product   of the vectors   ,   is orthogonal to both u and v.

Accepted Answer

The answer of ​Show that the cross product  ...

Question 17

Find the cross product   of the vectors   ,   .

Accepted Answer

A)     
B)     
C)     
D)     
E)     
A)     
B)     
C)     
D)     
E)

Question 18

Determine whether the set of vectors   in R3 is orthogonal (but not orthonormal), orthonormal, or neither.

Accepted Answer

A)  neither 
B)  orthonormal 
C) orthogonal 
A)  neither 
B)  orthonormal 
C) orthogonal

Question 19

Find the angle  in radians between the vectors   . Round your answers to three decimal places.

Accepted Answer

A)     
B)     
C)     
D)    
E)     
A)     
B)     
C)     
D)    
E)

Question 20

Use the functions   in C[-1, 1] to find   for the inner product   .

Accepted Answer

A)     
B)    
C)    
D)     
E)     
A)     
B)    
C)    
D)     
E)

Find bases for the four fundamental subspaces of the matrix .

Find an orthonormal basis for the solution space of the following homogeneous system of linear equations.

Find ,where .

S is the subspace of R³ consisting of the xz-plane. Find the orthogonal complement S.

Which of the following is a description of the orthogonal complement of

Find the distance d between u= and v= .

Find the least squares regression line for the points .

Find for the inner product defined in R³, where .

Find the coordinates of relative to the orthonormal basis in R³.

Use the Gram-Schmidt process to change the basis into an orthonormal basis for the subspace of R³ spanned by the vector .

Find the vector v with length 2 and the same direction as the vector .

Use the functions in C[-1, 1] to find for the inner product .

Find the length of the vector

Find for the inner product defined in R², where .

Use the functions in C[-1, 1] to find for the inner product .

Show that the cross product of the vectors , is orthogonal to both u and v.

Find the cross product of the vectors , .

Determine whether the set of vectors in R³ is orthogonal (but not orthonormal), orthonormal, or neither.

Find the angle in radians between the vectors . Round your answers to three decimal places.

Use the functions in C[-1, 1] to find for the inner product .

Linear Equations

Matrices

Determinants

Vector Spaces

Linear Transformations

Eigenvalues Eigenvectors

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Exam 5: Inner Product Spaces

Find bases for the four fundamental subspaces of the matrix .

Find an orthonormal basis for the solution space of the following homogeneous system of linear equations.

Find ,where .

S is the subspace of R3 consisting of the xz-plane. Find the orthogonal complement S.

Which of the following is a description of the orthogonal complement of

Find the distance d between u= and v= .

Find the least squares regression line for the points .

Find for the inner product defined in R3, where .

Find the coordinates of relative to the orthonormal basis in R3.

Use the Gram-Schmidt process to change the basis into an orthonormal basis for the subspace of R3 spanned by the vector .

Find the vector v with length 2 and the same direction as the vector .

Use the functions in C[-1, 1] to find for the inner product .

Find the length of the vector

Find for the inner product defined in R2, where .

Use the functions in C[-1, 1] to find for the inner product .

​Show that the cross product of the vectors , is orthogonal to both u and v.

Find the cross product of the vectors , .

Determine whether the set of vectors in R3 is orthogonal (but not orthonormal), orthonormal, or neither.

Find the angle in radians between the vectors . Round your answers to three decimal places.

Use the functions in C[-1, 1] to find for the inner product .

Linear Equations

Matrices

Determinants

Vector Spaces

Linear Transformations

Eigenvalues Eigenvectors

Filters

S is the subspace of R³ consisting of the xz-plane. Find the orthogonal complement S.

Find for the inner product defined in R³, where .

Find the coordinates of relative to the orthonormal basis in R³.

Use the Gram-Schmidt process to change the basis into an orthonormal basis for the subspace of R³ spanned by the vector .

Find for the inner product defined in R², where .

Show that the cross product of the vectors , is orthogonal to both u and v.

Determine whether the set of vectors in R³ is orthogonal (but not orthonormal), orthonormal, or neither.