Linear Space

Definition

Let $\mathcal{V}$ : Nonempty set of equally sized matrices

If

1. $\mathbf{A}+\mathbf{B}\in \mathcal{V},\forall \mathbf{A},\mathbf{B}\in \mathcal{V}$
2. $k\mathbf{A}\in \mathcal{V},\forall \mathbf{A}\in \mathcal{V},\forall k\in \mathbb{R}$

Then $\mathcal{V}$ is a linear space

Alternative definition

$$\forall {\mathbf{A}}_1,{\mathbf{A}}_2,\dots ,{\mathbf{A}}_m\in \mathcal{V},\forall k_1,k_2,\dots ,k_m\in \mathbb{R}⟹k_1{\mathbf{A}}_1+k_2{\mathbf{A}}_2+\cdots +k_m{\mathbf{A}}_m\in \mathcal{V}$$