Thread: Linear Algebra

1. Linear Algebra

My text is not helping at all and the wikipedia explanations of the concepts: nullspace and range are way over my head.

Given A is a 3x3 matrix
0 1 0
0 0 1
0 0 0

and B is also a 3x3 matrix
0 0 0
1 0 0
0 1 0

specify the nullspace of A and B, and the range of A and B

Thanks

2. Originally Posted by jblorien
My text is not helping at all and the wikipedia explanations of the concepts: nullspace and range are way over my head.

Given A is a 3x3 matrix
0 1 0
0 0 1
0 0 0

and B is also a 3x3 matrix
0 0 0
1 0 0
0 1 0

specify the nullspace of A and B, and the range of A and B

Thanks
The range is the span of the column vectors

$\displaystyle a \cdot \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} + b \cdot \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} + c \cdot \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix}$

These vectors span the xy plane so that is your range.

The null space is the solution to Ax=0

$\displaystyle \begin{bmatrix} 0 && 1 && 0 \\ 0 && 0 && 1 \\ 0 && 0 && 0 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$

using the augmented matrix we get

$\displaystyle \begin{bmatrix} 0 && 1 && 0 && 0 \\ 0 && 0 && 1 && 0\\ 0 && 0 && 0 && 0\\ \end{bmatrix}$

So by parameterizing our solution we

$\displaystyle x=t$
$\displaystyle y=0$
$\displaystyle z=0$

$\displaystyle \begin{bmatrix} t \\ 0 \\ 0 \\ \end{bmatrix} = t \cdot \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}$

So the null space is all of the points on the x axis.

3. Note that elementry row operations will change the column space of a matrix.

the span of the columns is

$\displaystyle a \cdot \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix} +b \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix} +c \cdot \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$

This is the yz plane.

Note that elementry row operation will NOT change the null space.

solving the augmented matrix

$\displaystyle \begin{bmatrix} 0 && 0 && 0 && 0\\ 1 && 0 && 0 && 0\\ 0 && 1 && 0 && 0\\ \end{bmatrix}$

in reduced row form

$\displaystyle \begin{bmatrix} 1 && 0 && 0 && 0\\ 0 && 1 && 0 && 0\\ 0 && 0 && 0 && 0\\ \end{bmatrix}$

so parameterizing the solution

$\displaystyle x=0$
$\displaystyle y=0$
$\displaystyle z=t$

so we get...

$\displaystyle \begin{bmatrix} 0 \\ 0 \\ t \\ \end{bmatrix} =t \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix}$

So the null space is all points on the z-axis.