MATH 01210 · Vector spaces

Subspaces

Definition

A subspace of a vector space V is a subset H closed under addition and scalar multiplication (and contains 0). Equivalently: closed under linear combinations.

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Examples: null space of A, column space of A, span of a set of vectors.

Lines through origin in R² are subspaces; lines not through origin are not.

To prove subspace: show nonempty/0, closed under + and scalar ·.

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Attempt each problem first, then open the worked solution.

1.Is {(x,y)|y=1} a subspace of R²? Why/why not?

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