# Thread: Linear independence of matrix set?

1. ## Linear independence of matrix set?

I need to show that the following set of 2x2 matrices forms a basis for the subspace that it spans, ie) that it is linearly independent.

{[1 1] [0 1] [2 0] }
{[1 0], [1 1], [0 -1]}

For a vector set I would just form a columnspace matrix, row reduce, and ensure that there are no parameters (that the rank is equal to the amount of variables).

How do I go about forming a matrix to use this method with the given 2x2 matrices.
Or, what method should I use to show that this matrix set is linearly independent??

2. Originally Posted by crymorenoobs
I need to show that the following set of 2x2 matrices forms a basis for the subspace that it spans, ie) that it is linearly independent.

{[1 1] [0 1] [2 0] }
{[1 0], [1 1], [0 -1]}

For a vector set I would just form a columnspace matrix, row reduce, and ensure that there are no parameters (that the rank is equal to the amount of variables).

How do I go about forming a matrix to use this method with the given 2x2 matrices.
Or, what method should I use to show that this matrix set is linearly independent??

Suppose $\displaystyle a\begin{pmatrix}1&1\\1&0\end{pmatrix}+b\begin{pmat rix}0&1\\1&1\end{pmatrix}+c\begin{pmatrix}2&0\\0&\ !\!\!-1\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix }$ , $\displaystyle a,b,c\in \mathbb{R}$ (in case you meant real matrices), then:

$\displaystyle \begin{pmatrix}a+2c&a+b\\a+b&b-c\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix }$

Well, now just compare entry-entry in both sides and conclude that the only solution is $\displaystyle a=b=c=0\Longrightarrow$ your matrices are lin. ind.

Tonio

3. Ah kk thanks.