Does a set of three matrices span the space of 2×2 matrices?

[sciencegem]

Hi all,

This isn't actually part of my assigned homework, I was just trying it out as the topic confuses me. I think I might understand what's going on a little more if someone could walk me through this. Any advice on the intuition behind it would be great. Thanks so much.

1. Homework Statement

Sorry I'm not sure how to input matrices properly...I've attached a pic off the web. It's question 11.

Homework Equations

I think the fundamental idea here is spanning sets, which is all the possible linear combinations of those matrices, right?

The Attempt at a Solution

[/B]
The truth is I'm not really sure what the logic behind solving this is. My attempt involved row reducing a matrix with A_1, A_2, and A_3 straightened out as it's columns, but I wasn't really sure what to do with the result as honestly I'm pretty slow and I don't have any intuition regarding this question. If someone could walk me through this I'd be extremely grateful :)

 

[sciencegem]

Figured it out :P just wasn't thinking it through at all. Sorry about that. If anyone has any gems they want to throw in about the nature of spanning sets or what not fantastic, otherwise case closed.

 

[HallsofIvy]

The span of a set of vectors is just their most general linear combination. The span of $\{v_1, v_2, \cdot\cdot\cdot, v_n\}$ is the the set of all vectors of the form $a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n$ where $a_1$, $a_2$, ..., $a_n$ can be any numbers. In problem 9, you are given that $A_1= \begin{pmatrix}1 & 2 \\ - 1 & 1\end{pmatrix}$ and $A_2= \begin{pmatrix}0 & 1 \\ 2 & 1 \end{pmatrix}$. The span of $\{A_1, A_2\}$ is the set of all matrices of the form
$$aA_1+ bA_2= a\begin{pmatrix}1 & 2 \\ - 1 & 1\end{pmatrix}+ b\begin{pmatrix}0 & 1 \\ 2 & 1 \end{pmatrix}= \begin{pmatrix}a & 2a+ b \\ 2b- a & a+ b \end{pmatrix} $$ where a and b are any two numbers.