Finite Sum of Vector Subspaces is a Vector Subspace

Last updated Nov 1, 2022

Let $V$ be a Vector Space over Field $F$ and suppose $W_{1}, \dots, W_{n} \subset V$ are Vector Subspaces of $V$ for some $n \in \mathbb{N}$. Then $$\sum\limits_{i=1}^{n} W_{i} := {\sum\limits_{i=1}^{n}\mathbf{x}{i} : \mathbf{x}{i} \in W_{i} \text{ for } i \in [n]}$$ is a Vector Subspace of $V$.

Let $\mathbf{a}, \mathbf{b} \in \sum\limits_{i=1}^{n} W_{i}$ and let $c \in F$. Then there exists $\mathbf{x}{i}, \mathbf{y}{i} \in W_{i}$ for $i in [n]$ so that $$\begin{align*} \mathbf{a} &= \mathbf{x}{1} + \cdots + \mathbf{x}{n}\\ \mathbf{b} &= \mathbf{y}{1} + \cdots + \mathbf{y}{n}. \end{align*}$$ Observe that $$\begin{align*} c \mathbf{a} + \mathbf{b} &= c (\mathbf{x}{1} + \cdots + \mathbf{x}{n}) + \mathbf{y}{1} + \cdots + \mathbf{y}{n}\\ &=(c \mathbf{x}{1} + \mathbf{y}{1}) + \cdots + (c \mathbf{x}{n} + \mathbf{y}{n}). \end{align*}$$ We know that $c \mathbf{x}{i} + \mathbf{y}{i} \in W_{i}$ for each $i \in [n]$ because A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition. Thus, by definition, we have that $c \mathbf{a} + \mathbf{b} \in \sum\limits_{i=1}^{n} W_{i}$. Because A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition, $\sum\limits_{i=1}^{n} W_{i}$ is a Vector Subspace of $V$. $\blacksquare$