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 4PS
To determine

(a)

A matrix with column space contains [110],[001], row space contains [12],[25]

The required matrix is [101001]

Given:

Column space is [110],[001]and row space is [12],[25].

Calculation:

Calculate the column numbers and row numbers individually,

1.C1+0.C2=(1,1,0)0.C1+1.C2=(0,0,1)0.R1+2.R2+5.R3=(2,5)0.R1+1.R2+2.R3=(1,2)

Therefore, the required matrix formed is [101001].

(b)

A matrix with column space has basis [113], nullspace has basis [311].

The required matrix would be impossible to create because the value of addition of dimensions is not 3.

Given:

Column space has basis [113], nullspace has basis [311].

Calculation:

To construct a matrix, the dimension of the column space and the dimension of the nullspace must sum up to 3, which is not possible with the column space has basis [113]and nullspace has basis [311].

Therefore, the required matrix is not possible.

(c)

A matrix with dimension of nullspace =1+ dimension of left nullspace.

The required matrix is [11]or[112112].

Given:

Dimension of nullspace =1+ dimension of left nullspace.

Calculation:

The dimension of nullspace is one more than the dimension of left nullspace.

Therefore, the required matrix could be formed as [11]or[112112].

(d)

A matrix with nullspace contains [13] and column space contains [31]

The required matrix is [9331].

Given:

Nullspace contains [13] and column space contains [31]

Calculation:

The left nullspace contains [13] and column space contains [31]

Therefore, the matrix formed is the required matrix is [9331]

(e)

A matrix when Row space = column space, but nullspace left nullspace.

The required matrix is not possible.

Given:

Row space = column space, but nullspace left nullspace

Calculation:

Row space = column space, that means m=n.

Consider m=nand then the nullspaces will have the same dimension

   mr=nr

However, N(A)andN(AT)are orthogonal to the row space and column space respectively and we can see that the condition here is opposite.

Therefore, the required matrix is not possible.

To determine

(b)

A matrix with column space has basis [113], nullspace has basis [311].

The required matrix would be impossible to create because the value of addition of dimensions is not 3.

Given:

Column space has basis [113], nullspace has basis [311].

Calculation:

To construct a matrix, the dimension of the column space and the dimension of the nullspace must sum up to 3, which is not possible with the column space has basis [113]and nullspace has basis [311].

Therefore, the required matrix is not possible.

(c)

A matrix with dimension of nullspace =1+ dimension of left nullspace.

The required matrix is [11]or[112112].

Given:

Dimension of nullspace =1+ dimension of left nullspace.

Calculation:

The dimension of nullspace is one more than the dimension of left nullspace.

Therefore, the required matrix could be formed as [11]or[112112].

(d)

A matrix with nullspace contains [13] and column space contains [31]

The required matrix is [9331].

Given:

Nullspace contains [13] and column space contains [31]

Calculation:

The left nullspace contains [13] and column space contains [31]

Therefore, the matrix formed is the required matrix is [9331]

(e)

A matrix when Row space = column space, but nullspace left nullspace.

The required matrix is not possible.

Given:

Row space = column space, but nullspace left nullspace

Calculation:

Row space = column space, that means m=n.

Consider m=nand then the nullspaces will have the same dimension

   mr=nr

However, N(A)andN(AT)are orthogonal to the row space and column space respectively and we can see that the condition here is opposite.

Therefore, the required matrix is not possible.

To determine

(c)

A matrix with dimension of nullspace =1+ dimension of left nullspace.

The required matrix is [11]or[112112].

Given:

Dimension of nullspace =1+ dimension of left nullspace.

Calculation:

The dimension of nullspace is one more than the dimension of left nullspace.

Therefore, the required matrix could be formed as [11]or[112112].

(d)

A matrix with nullspace contains [13] and column space contains [31]

The required matrix is [9331].

Given:

Nullspace contains [13] and column space contains [31]

Calculation:

The left nullspace contains [13] and column space contains [31]

Therefore, the matrix formed is the required matrix is [9331]

(e)

A matrix when Row space = column space, but nullspace left nullspace.

The required matrix is not possible.

Given:

Row space = column space, but nullspace left nullspace

Calculation:

Row space = column space, that means m=n.

Consider m=nand then the nullspaces will have the same dimension

   mr=nr

However, N(A)andN(AT)are orthogonal to the row space and column space respectively and we can see that the condition here is opposite.

Therefore, the required matrix is not possible.

To determine

(d)

A matrix with nullspace contains [13] and column space contains [31]

The required matrix is [9331].

Given:

Nullspace contains [13] and column space contains [31]

Calculation:

The left nullspace contains [13] and column space contains [31]

Therefore, the matrix formed is the required matrix is [9331]

(e)

A matrix when Row space = column space, but nullspace left nullspace.

The required matrix is not possible.

Given:

Row space = column space, but nullspace left nullspace

Calculation:

Row space = column space, that means m=n.

Consider m=nand then the nullspaces will have the same dimension

   mr=nr

However, N(A)andN(AT)are orthogonal to the row space and column space respectively and we can see that the condition here is opposite.

Therefore, the required matrix is not possible.

To determine

(e)

A matrix when Row space = column space, but nullspace left nullspace.

The required matrix is not possible.

Given:

Row space = column space, but nullspace left nullspace

Calculation:

Row space = column space, that means m=n.

Consider m=nand then the nullspaces will have the same dimension

   mr=nr

However, N(A)andN(AT)are orthogonal to the row space and column space respectively and we can see that the condition here is opposite.

Therefore, the required matrix is not possible.

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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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