Linear Algebra with Applications, 10th Edition

3.2 Subspaces

Given a vector space V, it is often possible to form another vector space by taking a subset S of V and using the operations of V. Since V is a vector space, the operations of addition and scalar multiplication always produce another vector in V. For a new system using a subset S of V as its universal set to be a vector space, the set S must be closed under the operations of addition and scalar multiplication. That is, the sum of two elements of S must always be an element of S, and the product of a scalar and an element of S must always be an element of S.

Example 1

Let

$$S=\left\{\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\right|x_2=2x_1\}$$

S is a subset of ${ℝ}^2$. If

$$\mathbf{x}=\left[\begin{array}{c}c\\ 2c\end{array}\right]$$

is any element of S and $\alpha$ is any scalar, then

$$\alpha \mathbf{x}=\alpha \left[\begin{array}{c}c\\ 2c\end{array}\right]=\left[\begin{array}{c}\alpha c\\ 2\alpha c\end{array}\right]$$

is also an element of S. If

$$\begin{array}{c}\left[\begin{array}{c}a\\ 2a\end{array}\right]\end{array}$$