# General Vector Spaces

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Question: Let W be the space spanned by f = sinx and g = cosx.

(a) Show that for any value of theta, f, = sin(x + theta) and g_1 = cos (x + theta) vectors in W.

(b) Show that f_1 and g_1 form a basis for W.

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#### Solution Preview

Because f = sin x and g = cos x span the vector space S, any linear or multiplicative combination of these two vectors will also be vectors in W.

Consider u=sin(x+theta) and v=cos(x+theta)

By trigonometric identities,

Then, u = sinx ...

#### Solution Summary

In about 180 words, this solution discusses the two proofs regarding vector spaces. The solution is brief, but concise and will be understandable for anyone with a basic understanding of this subject material.

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