4. (10pts) Let V = C3 be equipped with the standard inner product ＜, ＞. Let f: V → C be the unique linear map defined by In other words, f is an element in V*. Find the unique vector z V such that

8. (10pts) Let V and W be finite dimensional vector spaces and let T : V→ W be a linear map. Recall that there exists a unique linear map Tt : W*→ V*, called the transpose of T, where V* and W* are the dual spaces of V and W, respectively. Consider V = R2 and W = R3. Let T : V → W be the linear map defined by T(a,b)=(3a+6,a-26,2a-6) Then there exists a unique linear map Tt : W* →V* as above. Define by Hint: you need to either explicitly write down Tt(θ)(a,b) for any (a,b), or simply write down a 2 x 1 matris representing Tt(θ).

4. (10pts) Let V = C³ be equipped with the standard inner product ＜, ＞. Let f: V → C be the unique linear map defined by

In other words, f is an element in V*. Find the unique vector z V such that

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