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Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a linearly independent set,...

Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W = span{w⃗1,w⃗2,w⃗3}.

(a) Is there a linear transformation P : V → W such that P(⃗vi) = w⃗i for i = 1, 2, 3?

(b) Is there a linear transformation Q : W → V such that Q(w⃗i) = ⃗vi for i = 1, 2, 3?

Hint: the easiest way to show a linear transformation exists is to define a particular linear transformation. To define a linear transformation, you must specify what it does to every element in its domain.

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