Thread: Basis for the Null Space of a Matrix

1. Basis for the Null Space of a Matrix

"Let A = \begin{pmatrix}-1&2&-13&-6\\ 2&1&1&-3\\ -3&0&-9&0\\ 7&2&11&-6\end{pmatrix}

Determine a basis for the null space of A."

So I reduced it and got the following matrix:

\begin{pmatrix}1&0&3&0\\ 0&1&-5&-3\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}

Writing it in terms of parameters I got:

x1 = -3x3

x2 = 5x3 + 3x4

So I thought the vectors were:

\begin{pmatrix}-3\\ 5\\ 0\\ 0\end{pmatrix} and \begin{pmatrix}0\\ 3\\ 0\\ 0\end{pmatrix}

But this incorrect.

Any help?

2. Originally Posted by sktsasus
"Let A be the following matrix:

. . . . .$A\, =\, \begin{pmatrix}-1&2&-13&-6\\ 2&1&1&-3\\ -3&0&-9&0\\ 7&2&11&-6\end{pmatrix}$

Determine a basis for the null space of A."

So I reduced it and got the following matrix:

. . . . .$\begin{pmatrix}1&0&3&0\\ 0&1&-5&-3\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}$
I'm not getting this matrix. However, even if I had:

Originally Posted by sktsasus
Writing it in terms of parameters I got:

. . . . .$x_1\, =\, -3x_3$

. . . . .$x_2\, =\, 5x_3\, +\, 3x_4$

So I thought the vectors were:

. . . . .$v_1\, =\, \begin{pmatrix}-3\\ 5\\ 0\\ 0\end{pmatrix} \, \mbox{ and }\, v_2\, =\, \begin{pmatrix}0\\ 3\\ 0\\ 0\end{pmatrix}$

But this is incorrect. Any help?
If these vectors were correct then, since you've fixed x_3 = 0 and x_4 = 0, then you must necessarily have x_1 = 0 and x_2 = 0. I don't think this will be correct...?

Instead, try using the method outlined and illustrated here. If you don't get the correct answer then, since we cannot check work that we cannot see, please reply showing all of your steps, so we can try to figure out where things are going sideways. Thank you!

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