Chapter 3.5, Problem 2PS

To determine

The bases and dimensions for the four subspaces associated with A and B.

$A=\left[\begin{array}{l}124\\ 248\end{array}\right]$and $B=\left[\begin{array}{l}124\\ 258\end{array}\right]$.

For A,

The row space basis is $\text{row}1=\left(1,2,4\right)$, nullspace is $\left(-2,1,0\right)$and $\left(-4,0,1\right)$

The column space basis is $\text{column}1=\left(1,2\right)$, left nullspace is $\left(-2,1\right)$

For B,

The row space basis both rows are $\left(1,2,4\right)\text{and}\left(2,5,8\right)$, nullspace is $\left(-4,0,1\right)$

The column space basis two columns are $\left(1,2\right)\text{and}\left(2,5\right)$, left nullspace basis is empty.

Given:

$A=\left[\begin{array}{l}124\\ 248\end{array}\right]$and $B=\left[\begin{array}{l}124\\ 258\end{array}\right]$

Calculation:

Consider matrices $A=\left[\begin{array}{l}124\\ 248\end{array}\right]$and $B=\left[\begin{array}{l}124\\ 258\end{array}\right]$

For A, the row space basis is $\text{row}1=\left(1,2,4\right)$, nullspace is $\left(-2,1,0\right)$and $\left(-4,0,1\right)$, column space basis is $\text{column}1=\left(1,2\right)$, left nullspace is $\left(-2,1\right)$

For B, the row space basis both rows is $\left(1,2,4\right)\text{and}\left(2,5,8\right)$, nullspace is $\left(-4,0,1\right)$, the column space basis two columns is $\left(1,2\right)\text{and}\left(2,5\right)$, left nullspace basis is empty as the space contains only $y=0$that means the rows of B are independent.