# Why can't I prove this set of vectors spans

Mathematics Asked by Future Math person on February 18, 2021

$$begin{pmatrix} 1 & 1\ 1 & -1 end{pmatrix}$$,
$$begin{pmatrix} 0 & 2\ 3 & 0 end{pmatrix}$$, $$begin{pmatrix} 2 & 1\ -3 & -2 end{pmatrix}$$, $$begin{pmatrix} -1 & 4\ 5 & 1 end{pmatrix}$$

span $$M_2 (mathbb{R})$$.

I know that the traces of each of them are $$0$$ so they can’t possibly span $$M_2 (mathbb{R})$$ since you can’t write them as a linear combination with matrices that have non-zero traces.

However, I also attempted to do this with a system of equations. If I make a coefficient matrix from this, I get:

$$begin{pmatrix} 1 & 0 & 2 & -1 \ 1 & 2 & 1 & 4 \ 1 & 3 & -3 & 5 \ -1 & 0 & -2 & 1 \ end{pmatrix}$$.

After row reducing, I get:

$$begin{pmatrix} 1 & 0 & 0 & -13/7 \ 0& 1 & 0& 19/7 \ 0 & 0 & 1 & 3/7 \ 0 & 0 & 0 & 0 \ end{pmatrix}$$

this means the last variable is free and I have a consistent solution and hence, should span $$M_2 (mathbb{R})$$ but it doesn’t.

Why is my reasoning wrong?

As the last row is equal to zero, you won't be able to generate a vector having the last coordinate not equal to zero.

Hence those four matrices can't span $$M_2(mathbb R)$$.

Correct answer by mathcounterexamples.net on February 18, 2021