Chapter 3.5, Problem 4PS

To determine

(a)

A matrix with column space contains $\left[\begin{array}{l}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]$, row space contains $\left[\begin{array}{l}1\\ 2\end{array}\right],\left[\begin{array}{l}2\\ 5\end{array}\right]$

The required matrix is $\left[\begin{array}{l}10\\ 10\\ 01\end{array}\right]$

Given:

Column space is $\left[\begin{array}{l}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]$and row space is $\left[\begin{array}{l}1\\ 2\end{array}\right],\left[\begin{array}{l}2\\ 5\end{array}\right]$.

Calculation:

Calculate the column numbers and row numbers individually,

$\begin{array}{c}1.C_1+0.C_2=\left(1,1,0\right)\\ 0.C_1+1.C_2=\left(0,0,1\right)\\ 0.R_1+2.R_2+5.R_3=\left(2,5\right)\\ 0.R_1+1.R_2+2.R_3=\left(1,2\right)\end{array}$

Therefore, the required matrix formed is $\left[\begin{array}{l}10\\ 10\\ 01\end{array}\right]$.

(b)

A matrix with column space has basis $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$, nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

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

Given:

Column space has basis $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$, nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

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 $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$and nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

Therefore, the required matrix is not possible.

(c)

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

The required matrix is $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

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 $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

(d)

A matrix with nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

The required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$.

Given:

Nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Calculation:

The left nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Therefore, the matrix formed is the required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$

(e)

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

The required matrix is not possible.

Given:

Row space $=$ column space, but nullspace $\ne$ left nullspace

Calculation:

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

Consider $m=n$and then the nullspaces will have the same dimension

   $m-r=n-r$

However, $N\left(A\right)\text{and}N\left(A^{\text{T}}\right)$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 $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$, nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

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

Given:

Column space has basis $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$, nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

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 $\left[\begin{array}{l}1\\ 1\\ 3\end{array}\right]$and nullspace has basis $\left[\begin{array}{l}3\\ 1\\ 1\end{array}\right]$.

Therefore, the required matrix is not possible.

(c)

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

The required matrix is $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

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 $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

(d)

A matrix with nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

The required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$.

Given:

Nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Calculation:

The left nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Therefore, the matrix formed is the required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$

(e)

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

The required matrix is not possible.

Given:

Row space $=$ column space, but nullspace $\ne$ left nullspace

Calculation:

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

Consider $m=n$and then the nullspaces will have the same dimension

   $m-r=n-r$

However, $N\left(A\right)\text{and}N\left(A^{\text{T}}\right)$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 $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

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 $\left[11\right]\text{or}\left[\begin{array}{l}112\\ 112\end{array}\right]$.

(d)

A matrix with nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

The required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$.

Given:

Nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Calculation:

The left nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Therefore, the matrix formed is the required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$

(e)

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

The required matrix is not possible.

Given:

Row space $=$ column space, but nullspace $\ne$ left nullspace

Calculation:

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

Consider $m=n$and then the nullspaces will have the same dimension

   $m-r=n-r$

However, $N\left(A\right)\text{and}N\left(A^{\text{T}}\right)$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 $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

The required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$.

Given:

Nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Calculation:

The left nullspace contains $\left[\begin{array}{l}1\\ 3\end{array}\right]$ and column space contains $\left[\begin{array}{l}3\\ 1\end{array}\right]$

Therefore, the matrix formed is the required matrix is $\left[\begin{array}{l}-93\\ 3-1\end{array}\right]$

(e)

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

The required matrix is not possible.

Given:

Row space $=$ column space, but nullspace $\ne$ left nullspace

Calculation:

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

Consider $m=n$and then the nullspaces will have the same dimension

   $m-r=n-r$

However, $N\left(A\right)\text{and}N\left(A^{\text{T}}\right)$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 $\ne$ left nullspace.

The required matrix is not possible.

Given:

Row space $=$ column space, but nullspace $\ne$ left nullspace

Calculation:

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

Consider $m=n$and then the nullspaces will have the same dimension

   $m-r=n-r$

However, $N\left(A\right)\text{and}N\left(A^{\text{T}}\right)$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.