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Let v = (v1, · · · , vn), w = (w1, · · · ,...

Let v = (v1, · · · , vn), w = (w1, · · · , wn) ? R^n and let <v, w> denote the inner product on R n given by <v, w>= v1w1 + · · · + vnwn. Prove that for any linear transformation T : R^n ? R, there exists a fixed vector v ? R^n such that T(w) = <v, w>

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Answer #1

Let v = (v1, · · · , vn) be a fixed vector? Rn and let w = (w1, · · · , wn) ? Rn be an arbitrary vector. Further, let us define T: Rn ? R by T(w) = < v,w >. We will show that T is a linear transformation.

Let w = (w1, · · · , wn) and u = (u1, · · · , un) be 2 arbitrary vectors ? Rn and let k be an arbitrary scalar. Then T(w+u)= <v,w+u> = v1(w1+u1) + · · · + vn (wn+un) = v1w1 + · · · + vnwn + v1 u1 + · · · + vn un = < v,w > +< v,u > = T(w)+T(u). This means that T preserves vector addition. Also, T(kw) =<v,kw > = v1kw1 + · · · + vn kwn= k(v1w1 + · · · + vnwn) = kT(w). This means that T preserves scalar multiplication. Hence T is a linear transformation.

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