<s>[INST] Finding values to make a set a basis Question:
For which values of $a\in \Bbb R$ is the set below a basis for $M_{2x2}(\Bbb R)$ as a vector space over $\Bbb R$?
$${(a,2a,2,3a), (1,2,2a,3), (1,2a,a+1,a+2), (1,a+1,2,2a+1)}$$ (these vectors should be matrices but I do not know how to do them in MathJax)
My thinking is that in order to be a basis the set must be linearly independent. Thus the sum of the set of vectors and the set of scalar coefficients must equal zero where all scalar coefficients are equal to zero. If this is the case then any number could be $a$ without affecting the linear independence.
Is my thinking totally off? Can someone explain this problem if it is.
 [/INST] In order for the matrices presented to form a basis they need to be linearly independent. This means that for scalars $x_i$, the equation:
$$
  x_1 \begin{pmatrix} a & 2a \\ 2 & 3a \end{pmatrix}
+ x_2 \begin{pmatrix} 1 & 2 \\ 2a & 3 \end{pmatrix}
+ x_3 \begin{pmatrix} 1 & 2a \\ a+1 & a+2 \end{pmatrix}
+ x_4 \begin{pmatrix} 1 & a+1 \\ 2 & 2a+1 \end{pmatrix}
= \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
$$
should have only the trivial solution. To solve this, we need to solve the system:
$$
\begin{align}
ax_1 + x_2 + x_3 + x_4 &= 0 \\
2ax_1 + 2x_2 +2ax_3 +(a+1)x_4 &= 0 \\
2x_1 + 2ax_2 + (a+1)x_3 + 2x_4 &= 0 \\
3ax_1 +3x_2 +(a+2)x_3 + (2a+1)x_4 &= 0
\end{align}
$$
Ensuring that this system has only the trivial solution amounts to making sure that $\det{A} \neq 0$ where:
$$
A = \begin{pmatrix} a & 1 & 1 & 1 \\ 2a & 2 & 2a & a+1 \\ 2 & 2a & a+1 & 2 \\ 3a & 3 & a+2 &2a+1 \end{pmatrix}
$$
After finding the determinant of $A$ you can find which $a$ values result in $\det{A} \neq 0$ It is precisely these values of $a$ which result in the set being a basis.
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