The combination of rows of A that produced zero. Also, the vectors in the nullspace of A T and nullspace of A. The combination that gives zero rows is: ( R o w 3 ) − 2 × ( R o w 2 ) + ( R o w 1 ) The nullspace of A T and the nullspace of A is spanned by the vector ( 1 , − 2 , 1 ) . Given: [ A b ] = [ 1 2 3 b 1 4 5 6 b 2 7 8 9 b 3 ] → [ 1 − 2 3 b 1 0 − 3 − 6 b 2 − 4 b 1 0 0 0 b 3 − 2 b 2 + b 1 ] Calculation: Consider the matrix [ A b ] = [ 1 2 3 b 1 4 5 6 b 2 7 8 9 b 3 ] → [ 1 − 2 3 b 1 0 − 3 − 6 b 2 − 4 b 1 0 0 0 b 3 − 2 b 2 + b 1 ] The row that gives zero row is b 3 that means the combination is ( R o w 3 ) − 2 × ( R o w 2 ) + ( R o w 1 ) . As ( R o w 3 ) − 2 × ( R o w 2 ) + ( R o w 1 ) = 0 , this implies that the vectors c ( 1 , − 2 , 1 ) are present in the left nullspace. Thus, the combination is ( R o w 3 ) − 2 × ( R o w 2 ) + ( R o w 1 ) and the nullspace of A T and the nullspace of A is spanned by the vector ( 1 , − 2 , 1 ) .

Question

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Answer 1

Question

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

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Answer 2

Question

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

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Answer 3

Question

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

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Answer 4

Question

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

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Answer 5

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Answer 6

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Answer 7

Chapter 3.5, Problem 18PS

To determine

The combination of rows of A that produced zero. Also, the vectors in the nullspace of AT and nullspace of A.

The combination that gives zero rows is: (Row 3)−2×(Row 2)+(Row 1)

The nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Given:

[A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

Calculation:

Consider the matrix [A b] =[1 2 3 b14 5 6 b27 8 9 b3] → [1 −2 3 b10 −3 −6 b2−4b10 0 0 b3−2b2+b1]

The row that gives zero row is b3that means the combination is (Row 3)−2×(Row 2)+(Row 1).

As (Row 3)−2×(Row 2)+(Row 1)=0, this implies that the vectors c(1, −2, 1)are present in the left nullspace.

Thus, the combination is (Row 3)−2×(Row 2)+(Row 1)and the nullspace of AT and the nullspace of A is spanned by the vector (1, −2, 1).

Answer 8

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Answer 9

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Answer 10

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Answer 11

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

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