The Vector Subspace of 2 x 2 Matrices

Recall that the set of all $m \times n$ matrices denoted $M_{mn}$ forms a vector space, as verified on The Vector Space of m x n Matrices page. We will now begin to show that $M_{22}$, the set of all $2 \times 2$ matrices is a subspace of $M_{mn}$.

Recall that we only need to verify that the closure of addition and closure of scalar multiplication axioms hold, since we know that the other axioms are inherited since clearly $M_{22} \subset M_{mn}$.

Let $u, v \in M_{mn}$ such that $u = \begin{bmatrix} u_{11} & u_{12}\\ u_{21} & u_{22} \end{bmatrix}$ and $v = \begin{bmatrix} v_{11} & v_{12}\\ v_{21} & v_{22} \end{bmatrix}$, and let $a \in \mathbb{F}$

• 9. $u + v = \begin{bmatrix} u_{11} & u_{12}\\ u_{21} & u_{22} \end{bmatrix} + \begin{bmatrix} v_{11} & v_{12}\\ v_{21} & v_{22} \end{bmatrix} = \begin{bmatrix} u_{11} + v_{11} & u_{12} + v_{12} \\ u_{21} + v_{21} & u_{22} + v_{22} \end{bmatrix}$. This is still a $2 \times 2$ matrix, and so $(u + v) \in M_{22}$ and thus $M_{22}$ is closed under matrix addition.

• 10. $au = a \begin{bmatrix} u_{11} & u_{12}\\ u_{21} & u_{22} \end{bmatrix} = \begin{bmatrix} au_{11} & au_{12}\\ au_{21} & au_{22} \end{bmatrix}$ which is still a $2 \times 2$ matrix, and so $(au) \in M_{22}$ and $M_{22}$ is closed under scalar multiplication.

Therefore since axioms 9 and 10 hold, the subset $M_{22} \subset M_{mn}$ is a vector space, namely a vector subspace of the vector space $M_{mn}$.

The Vector Subspace of 2 x 2 Diagonal Matrices

Let $W$ be the set of $2 \times 2$ diagonal matrices. We note that $W \subset M_{22} \subset M_{mn}$, and we will now verify that $W$ is a subspace of $M_{22}$ and $M_{mn}$.

Let $u, v \in W$ such that $u = \begin{bmatrix} u_{11} & 0 \\ 0 & u_{22} \end{bmatrix}$ and $v = \begin{bmatrix} v_{11} & 0 \\ 0 & v_{22} \end{bmatrix}$, and let $a \in \mathbb{F}$.

• 9. $u + v = \begin{bmatrix} u_{11} & 0 \\ 0 & u_{22} \end{bmatrix} + \begin{bmatrix} v_{11} & 0 \\ 0 & v_{22} \end{bmatrix} = \begin{bmatrix} u_{11} + v_{11} & 0 \\ 0 & u_{22} v_{22} \end{bmatrix}$ which is still a diagonal matrix, so $(u + v) \in W$, and so $W$ is closed under addition.

• 10. $au = a \begin{bmatrix} u_{11} & 0 \\ 0 & u_{22} \end{bmatrix} = \begin{bmatrix} au_{11} & 0 \\ 0 & au_{22} \end{bmatrix}$ which is still a diagonal matrix, so $(au) \in W$, and so $W$ is closed under scalar multiplication.