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={[x1x2]|x2=2x1}

S is a subset of 2. If

x=[c2c]

is any element of S and α is any scalar, then

αx=α[c2c]=[αc2αc]

is also an element of S. If

[a2a]

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