Question

Let W be the set of all 3×3 real diagonal matrices. Prove that W is a subspace of M33.

Answer 1

Let $A,B ∈ W$ and $α∈ℝ$ such that

$A=\begin{pmatrix} a_{11} & 0&0 \\ 0 & a_{22}&0\\0&0&a_{33} \end{pmatrix}$ $B=\begin{pmatrix} b_{11} & 0&0 \\ 0 & b_{22}&0\\0&0&b_{33} \end{pmatrix}$

$A+αB=\begin{pmatrix} a_{11} & 0&0 \\ 0 & a_{22}&0\\0&0&a_{33} \end{pmatrix} + α\begin{pmatrix} b_{11} & 0&0 \\ 0 & b_{22}&0\\0&0&b_{33} \end{pmatrix}$

$=\begin{pmatrix} a_{11} & 0&0 \\ 0 & a_{22}&0\\0&0&a_{33} \end{pmatrix} + \begin{pmatrix} αb_{11} & 0&0 \\ 0 & αb_{22}&0\\0&0&αb_{33} \end{pmatrix}$

$=\begin{pmatrix} a_{11}+αb_{11} & 0&0 \\ 0 & a_{22}+αb_{22}&0\\0&0&a_{33}+αb_{33} \end{pmatrix}$

Hence, $A+αB∈W$

Thus, $W$ is a subspace of $M_{3×3}(ℝ)$