Question

4. Prove the Following: a. Prove that if V is a vector space with subspace W...

4. Prove the Following:

a. Prove that if V is a vector space with subspace W ⊂ V, and if U ⊂ W is a subspace of the vector space W, then U is also a subspace of V

b. Given span of a finite collection of vectors {v1, . . . , vn} ⊂ V as follows:

Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in the scalar field} = {all linear combinations of the vi}

We can generalize this definition for any subset S ⊂ V: Span(S) := {all linear combinations of vectors in S}

Prove that in fact Span(S) is always a subspace of V when S is non-empty (i.e. it is true even if S is not necessarily finite).

c. Suppose {v1,. . . , vn} is a finite subset of a vector space V which is linearly independent.

i.Prove that for any i, {v1, . . . , vn} \ {vi} is also a linearly independent set (as long as n>1).

ii.Prove that if {v1, . . . , vn} is not a basis for V then you can find a vector w ∈ V such that {v1, . . . , vn, w} is a linearly independent set.

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