# 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* is a subset of ${\mathbb{R}}^{2}$. If

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

is also an element of *S*. If

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