Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
5th Edition
ISBN: 9780980232776
Author: Gilbert Strang
Publisher: Wellesley-Cambridge Press
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Chapter 3.5, Problem 21PS
To determine

(a)

The vectors that span the column space of A when A is the sum of two matrices of rank one: A=uvT+wzT

The column space of A is spanned by vectors u and w.

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Consider the equation A=uvT+wzT

We can clearly see that the vectors u and w span the column space of A.

Thus, the column space of A is spanned by vectors u and w.

(b)

To Calculate:

The vectors that span the row space of A when A is the sum of two matrices of rank one: A=uvT+wzT.

The row space of A is spanned by vectors v and z.

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Consider the equation A=uvT+wzT

We can clearly see that the vectors v and z span the row space of A.

Thus, the row space of A is spanned by vectors v and z.

(c) To Fill:

The blank space.

The rank is less than 2 if vectors u and w are collinear/dependent or if vectors v and z are collinear/dependent.

Given:

The rank is less than 2 if ____ or if _____.

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Collinear means that one vector is multiple of another and it is noted that two collinear vectors are always meant to be linearly dependent.

It is observed that u and w are multiples of each other and also v and z are collinear because they are also multiple of each other.

Therefore, the rank of matrix A is less than 2 only when the vectors u and w are dependent or when the vectors v and z are dependent.

(d)

A and its rank if u=z=(1,0,0)and v=w=(0,0,1)

The rank of A=uvT+wzT is 2

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

u=z=(1,0,0)and v=w=(0,0,1)

Calculation:

Substitute the values of u=z=(1,0,0)and v=w=(0,0,1) in equation A=uvT+wzT

A=uvT+wzT

A=[100]×[001]+[001]×[100]

A=[001000000]+[000000100]

A=[001000100]

Therefore, the rank of matrix A is r(A)=2

To determine

(b)

To Calculate:

The vectors that span the row space of A when A is the sum of two matrices of rank one: A=uvT+wzT.

The row space of A is spanned by vectors v and z.

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Consider the equation A=uvT+wzT

We can clearly see that the vectors v and z span the row space of A.

Thus, the row space of A is spanned by vectors v and z.

(c) To Fill:

The blank space.

The rank is less than 2 if vectors u and w are collinear/dependent or if vectors v and z are collinear/dependent.

Given:

The rank is less than 2 if ____ or if _____.

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Collinear means that one vector is multiple of another and it is noted that two collinear vectors are always meant to be linearly dependent.

It is observed that u and w are multiples of each other and also v and z are collinear because they are also multiple of each other.

Therefore, the rank of matrix A is less than 2 only when the vectors u and w are dependent or when the vectors v and z are dependent.

(d)

A and its rank if u=z=(1,0,0)and v=w=(0,0,1)

The rank of A=uvT+wzT is 2

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

u=z=(1,0,0)and v=w=(0,0,1)

Calculation:

Substitute the values of u=z=(1,0,0)and v=w=(0,0,1) in equation A=uvT+wzT

A=uvT+wzT

A=[100]×[001]+[001]×[100]

A=[001000000]+[000000100]

A=[001000100]

Therefore, the rank of matrix A is r(A)=2

To determine

(c) To Fill:

The blank space.

The rank is less than 2 if vectors u and w are collinear/dependent or if vectors v and z are collinear/dependent.

Given:

The rank is less than 2 if ____ or if _____.

Sum of two matrices of rank one is A: A=uvT+wzT

Calculation:

Collinear means that one vector is multiple of another and it is noted that two collinear vectors are always meant to be linearly dependent.

It is observed that u and w are multiples of each other and also v and z are collinear because they are also multiple of each other.

Therefore, the rank of matrix A is less than 2 only when the vectors u and w are dependent or when the vectors v and z are dependent.

(d)

A and its rank if u=z=(1,0,0)and v=w=(0,0,1)

The rank of A=uvT+wzT is 2

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

u=z=(1,0,0)and v=w=(0,0,1)

Calculation:

Substitute the values of u=z=(1,0,0)and v=w=(0,0,1) in equation A=uvT+wzT

A=uvT+wzT

A=[100]×[001]+[001]×[100]

A=[001000000]+[000000100]

A=[001000100]

Therefore, the rank of matrix A is r(A)=2

To determine

(d)

A and its rank if u=z=(1,0,0)and v=w=(0,0,1)

The rank of A=uvT+wzT is 2

Given:

Sum of two matrices of rank one is A: A=uvT+wzT

u=z=(1,0,0)and v=w=(0,0,1)

Calculation:

Substitute the values of u=z=(1,0,0)and v=w=(0,0,1) in equation A=uvT+wzT

A=uvT+wzT

A=[100]×[001]+[001]×[100]

A=[001000000]+[000000100]

A=[001000100]

Therefore, the rank of matrix A is r(A)=2

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Chapter 3 Solutions

Introduction to Linear Algebra, Fifth Edition

Ch. 3.1 - Prob. 11PSCh. 3.1 - Prob. 12PSCh. 3.1 - Prob. 13PSCh. 3.1 - Prob. 14PSCh. 3.1 - Prob. 15PSCh. 3.1 - Prob. 16PSCh. 3.1 - Prob. 17PSCh. 3.1 - Prob. 18PSCh. 3.1 - Prob. 19PSCh. 3.1 - Prob. 20PSCh. 3.1 - Prob. 21PSCh. 3.1 - Prob. 22PSCh. 3.1 - Prob. 23PSCh. 3.1 - Prob. 24PSCh. 3.1 - Prob. 25PSCh. 3.1 - Prob. 26PSCh. 3.1 - Prob. 27PSCh. 3.1 - Prob. 28PSCh. 3.1 - Prob. 29PSCh. 3.1 - Prob. 30PSCh. 3.1 - Prob. 31PSCh. 3.1 - Prob. 32PSCh. 3.2 - Prob. 1PSCh. 3.2 - Prob. 2PSCh. 3.2 - Prob. 3PSCh. 3.2 - Prob. 4PSCh. 3.2 - Prob. 5PSCh. 3.2 - Prob. 6PSCh. 3.2 - Prob. 7PSCh. 3.2 - Prob. 8PSCh. 3.2 - Prob. 9PSCh. 3.2 - Prob. 10PSCh. 3.2 - Prob. 11PSCh. 3.2 - Prob. 12PSCh. 3.2 - Prob. 13PSCh. 3.2 - Prob. 14PSCh. 3.2 - Prob. 15PSCh. 3.2 - Prob. 16PSCh. 3.2 - Prob. 17PSCh. 3.2 - Prob. 18PSCh. 3.2 - Prob. 19PSCh. 3.2 - Prob. 20PSCh. 3.2 - Prob. 21PSCh. 3.2 - Prob. 22PSCh. 3.2 - Prob. 23PSCh. 3.2 - Prob. 24PSCh. 3.2 - Prob. 25PSCh. 3.2 - Prob. 26PSCh. 3.2 - Prob. 27PSCh. 3.2 - Prob. 28PSCh. 3.2 - Prob. 29PSCh. 3.2 - Prob. 30PSCh. 3.2 - Prob. 31PSCh. 3.2 - Prob. 32PSCh. 3.2 - Prob. 33PSCh. 3.2 - Prob. 34PSCh. 3.2 - Prob. 35PSCh. 3.2 - Prob. 36PSCh. 3.2 - Prob. 37PSCh. 3.2 - Prob. 38PSCh. 3.2 - Prob. 39PSCh. 3.2 - Prob. 40PSCh. 3.2 - Prob. 41PSCh. 3.2 - Prob. 42PSCh. 3.2 - Prob. 43PSCh. 3.2 - Prob. 44PSCh. 3.2 - Prob. 45PSCh. 3.2 - Prob. 46PSCh. 3.2 - Prob. 47PSCh. 3.2 - Prob. 48PSCh. 3.2 - Prob. 49PSCh. 3.2 - Prob. 50PSCh. 3.2 - Prob. 51PSCh. 3.2 - Prob. 52PSCh. 3.2 - Prob. 53PSCh. 3.2 - Prob. 54PSCh. 3.2 - Prob. 55PSCh. 3.2 - Prob. 56PSCh. 3.2 - Prob. 57PSCh. 3.2 - Prob. 58PSCh. 3.2 - Prob. 59PSCh. 3.2 - Prob. 60PSCh. 3.3 - Prob. 1PSCh. 3.3 - Prob. 2PSCh. 3.3 - Prob. 3PSCh. 3.3 - Prob. 4PSCh. 3.3 - Prob. 5PSCh. 3.3 - Prob. 6PSCh. 3.3 - Prob. 7PSCh. 3.3 - Prob. 8PSCh. 3.3 - Prob. 9PSCh. 3.3 - Prob. 10PSCh. 3.3 - Prob. 11PSCh. 3.3 - Prob. 12PSCh. 3.3 - Prob. 13PSCh. 3.3 - Prob. 14PSCh. 3.3 - Prob. 15PSCh. 3.3 - Prob. 16PSCh. 3.3 - Prob. 17PSCh. 3.3 - Prob. 18PSCh. 3.3 - Prob. 19PSCh. 3.3 - Prob. 20PSCh. 3.3 - Prob. 21PSCh. 3.3 - Prob. 22PSCh. 3.3 - Prob. 23PSCh. 3.3 - Prob. 24PSCh. 3.3 - Prob. 25PSCh. 3.3 - Prob. 26PSCh. 3.3 - Prob. 27PSCh. 3.3 - Prob. 28PSCh. 3.3 - Prob. 29PSCh. 3.3 - Prob. 30PSCh. 3.3 - Prob. 31PSCh. 3.3 - Prob. 32PSCh. 3.3 - Prob. 33PSCh. 3.3 - Prob. 34PSCh. 3.3 - Prob. 35PSCh. 3.3 - Prob. 36PSCh. 3.3 - Prob. 37PSCh. 3.4 - Prob. 1PSCh. 3.4 - Prob. 2PSCh. 3.4 - Prob. 3PSCh. 3.4 - Prob. 4PSCh. 3.4 - Prob. 5PSCh. 3.4 - Prob. 6PSCh. 3.4 - Prob. 7PSCh. 3.4 - Prob. 8PSCh. 3.4 - Prob. 9PSCh. 3.4 - Prob. 10PSCh. 3.4 - Prob. 11PSCh. 3.4 - Prob. 12PSCh. 3.4 - Prob. 13PSCh. 3.4 - Prob. 14PSCh. 3.4 - Prob. 15PSCh. 3.4 - Prob. 16PSCh. 3.4 - Prob. 17PSCh. 3.4 - Prob. 18PSCh. 3.4 - Prob. 19PSCh. 3.4 - Prob. 20PSCh. 3.4 - Prob. 21PSCh. 3.4 - Prob. 22PSCh. 3.4 - Prob. 23PSCh. 3.4 - Prob. 24PSCh. 3.4 - Prob. 25PSCh. 3.4 - Prob. 26PSCh. 3.4 - Prob. 27PSCh. 3.4 - Prob. 28PSCh. 3.4 - Prob. 29PSCh. 3.4 - Prob. 30PSCh. 3.4 - Prob. 31PSCh. 3.4 - Prob. 32PSCh. 3.4 - Prob. 33PSCh. 3.4 - Prob. 34PSCh. 3.4 - Prob. 35PSCh. 3.4 - Prob. 36PSCh. 3.4 - Prob. 37PSCh. 3.4 - Prob. 38PSCh. 3.4 - Prob. 39PSCh. 3.4 - Prob. 40PSCh. 3.4 - Prob. 41PSCh. 3.4 - Prob. 42PSCh. 3.4 - Prob. 43PSCh. 3.4 - Prob. 44PSCh. 3.4 - Prob. 45PSCh. 3.4 - Prob. 46PSCh. 3.5 - Prob. 1PSCh. 3.5 - Prob. 2PSCh. 3.5 - Prob. 3PSCh. 3.5 - Prob. 4PSCh. 3.5 - Prob. 5PSCh. 3.5 - Prob. 6PSCh. 3.5 - Prob. 7PSCh. 3.5 - Prob. 8PSCh. 3.5 - Prob. 9PSCh. 3.5 - Prob. 10PSCh. 3.5 - Prob. 11PSCh. 3.5 - Prob. 12PSCh. 3.5 - Prob. 13PSCh. 3.5 - Prob. 14PSCh. 3.5 - Prob. 15PSCh. 3.5 - Prob. 16PSCh. 3.5 - Prob. 17PSCh. 3.5 - Prob. 18PSCh. 3.5 - Prob. 19PSCh. 3.5 - Prob. 20PSCh. 3.5 - Prob. 21PSCh. 3.5 - Prob. 22PSCh. 3.5 - Prob. 23PSCh. 3.5 - Prob. 24PSCh. 3.5 - Prob. 25PSCh. 3.5 - Prob. 26PSCh. 3.5 - Prob. 27PSCh. 3.5 - Prob. 28PSCh. 3.5 - Prob. 29PSCh. 3.5 - Prob. 30PSCh. 3.5 - Prob. 31PS
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