(a) The matrix that leads X to A X = zero matrix. A X = 0 when X is in the form of [ a b c a b c a b c ] or when each column of X is a multiple of ( 1 , 1 , 1 ) Given: M is the space of 3 by 3 matrices and multiply every matrix X in M by A = [ 1 0 − 1 − 1 1 0 0 − 1 1 ] . Notice A [ 1 1 1 ] = [ 0 0 0 ] Calculation: Consider row reduction method to check that the rows of A are dependent and A has rank 2. The nullspace of A has basis: [ 1 1 1 ] Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N ( A ) , then A X = 0 will have the form: [ a b c a b c a b c ] having dim ( nullspace ) = 3 . (b) The matrix that have the form AX for some matrix X. A X = B when all the columns of matrix add to zero : [ a b c d e f − a − d − b − e − c − f ] Given: M is the space of 3 by 3 matrices and multiply every matrix X in M by A = [ 1 0 − 1 − 1 1 0 0 − 1 1 ] . Notice A [ 1 1 1 ] = [ 0 0 0 ] Calculation: The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means, A X = B only when [ a b c d e f − a − d − b − e − c − f ] because adding these columns will result in zero and thus the dim ( B ) = 6 . (c) The dimensions of nullspace and column space of M. Also, reason behind ( n − r ) + r = 9 . dim ( B ) = 6 and dim ( nullspace ) = 3 ( n − r ) + r = 9 because there are 9 entries in the matrix Given: M is the space of 3 by 3 matrices and multiply every matrix X in M by A = [ 1 0 − 1 − 1 1 0 0 − 1 1 ] . Notice A [ 1 1 1 ] = [ 0 0 0 ] Calculation: It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6. Thus, dimension of a 3 by 3 matrix M will be 3 + 6 = 9 Therefore, dim ( M 3x3 ) = 9 which means there are 9 entries in the 3 by 3 matrix.

Question

Want to see the full answer?

Answer 1

Question

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

I want a mathematical relationship with all the details, not explanations and definitions

4 sinx cos2x+4 cos x sin2x-1=0

For the matrix A, find (if possible) a nonsingular matrix P such that P-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -2 -[47] A = -3 1 P = Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. P-1AP =

Answer 2

Question

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 6

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 7

Chapter 3.5, Problem 30PS

To determine

(a)

The matrix that leads X to AX=zero matrix.

AX=0when X is in the form of [a b ca b ca b c]or when each column of X is a multiple of (1, 1, 1)

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

Consider row reduction method to check that the rows of A are dependent and A has rank 2.

The nullspace of A has basis: [111]

Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of N(A), then AX=0 will have the form: [a b ca b ca b c]having dim(nullspace)=3.

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 8

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 9

To determine

(b)

The matrix that have the form AX for some matrix X.

AX=Bwhen all the columns of matrix add to zero : [ a b c d e f−a−d −b−e −c−f]

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,

AX=Bonly when [ a b c d e f−a−d −b−e −c−f]because adding these columns will result in zero and thus the dim(B)=6.

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 10

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 11

To determine

(c)

The dimensions of nullspace and column space of M. Also, reason behind (n−r)+r=9.

dim(B)=6and dim(nullspace)=3

(n−r)+r=9because there are 9 entries in the matrix

Given:

M is the space of 3 by 3 matrices and multiply every matrix X in M by

A=[ 1 0 −1−1 1 0 0 −1 1]. Notice A[111]=[000]

Calculation:

It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.

Thus, dimension of a 3 by 3 matrix M will be 3+6=9

Therefore, dim(M3x3)=9which means there are 9 entries in the 3 by 3 matrix.

Answer 12

Expert Solution & Answer

Answer 13

Expert Solution & Answer

Answer 14

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 15

Ch. 3.1 - Prob. 1PS Ch. 3.1 - Prob. 2PS Ch. 3.1 - Prob. 3PS Ch. 3.1 - Prob. 4PS Ch. 3.1 - Prob. 5PS Ch. 3.1 - Prob. 6PS Ch. 3.1 - Prob. 7PS Ch. 3.1 - Prob. 8PS Ch. 3.1 - Prob. 9PS Ch. 3.1 - Prob. 10PS

Videos

Want to see the full answer?

Chapter 3 Solutions