[Community Question] Linear-algebra: Pauli matrices span the set of $2 \times 2$ matrices over $\mathbb{C}$?
Define $B = \{\frac{1}{\sqrt{2}}I, \frac{1}{\sqrt{2}}X, \frac{1}{\sqrt{2}}Y, \frac{1}{\sqrt{2}}Z \}$, where:
$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\\end{bmatrix}$, $Y = \begin{bmatrix} 0 & i \\ -i & 0 \\ \end{bmatrix}$, and $Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}$
I am trying to show that B (modified from the Pauli matrices) span the set of $2 \times 2$ matrices with entries in $\mathbb{C}$ (denote $\mathcal{M}_2(\mathbb{C})$).
If $A \in \mathcal{M}_2(\mathbb{C})$, then we can write:
$A = \frac{1}{\sqrt{2}} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ for some $a, b, c, d \in \mathbb{C}$.
I think that I want to show that $A = \frac{1}{\sqrt{2}}(\lambda_0I + \lambda_1X + \lambda_2Y + \lambda_3Z)$?
This means I would need to show that the system given by: $\{a = \lambda_0 + \lambda_3, \ b = \lambda_1 + \lambda_2i, \ c = \lambda_1 - \lambda_2i, \ d = \lambda_0 - \lambda_3\}$ is always solvable. How do I go about doing this? Or should I approach the question some other way?
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