linear subspace

Definition

Consider field $\mathbb{F}$.

Let $(\mathcal{V}, \mathbb{F})$ be a vector space, and let $\mathcal{W} \subseteq \mathcal{V}$. Then, $(\mathcal{W}, \mathbb{F})$ is a subspace of $(\mathcal{V}, \mathbb{F})$ if and only if $(\mathcal{W}, \mathbb{F})$ is a vector space or, equivalently, if and only if $(\alpha w_1 + \beta w_2) \in \mathcal{W}$ for all $\alpha, \beta \in \mathbb{F}$, and for all $w_1, w_2 \in \mathcal{W}$.

References

1. A. J. Laub, Matrix Analysis for Scientists and Engineers, Society for Industrial and Applied Mathematics, 2005, pp. 9-10.
2. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, vol. 2, 2 vols. Graylock Press, 1961.