4. (8 points) Let V and W be vector spaces over R and T: VW a...

 

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4. (8 points) Let V and W be vector spaces over R and T: VW a linear transforma- tion. Let {₁,..., Un} be a basis for V. (a) Prove that {T(v₁),..., T(vn)} is a spanning set for range(T). (In your argument, indicate clearly where you are using the facts that (i) {v₁,..., Un} is a basis for V and (ii) T is linear.) (b) Give a concrete example of vector spaces V and W, a basis {₁,..., Un} of V, and linear transformation T such that {T(v₁),..., T(vn)} is not a basis for range(T). 4. (8 points) Let V and W be vector spaces over R and T: VW a linear transforma- tion. Let {₁,..., Un} be a basis for V. (a) Prove that {T(v₁),..., T(vn)} is a spanning set for range(T). (In your argument, indicate clearly where you are using the facts that (i) {v₁,..., Un} is a basis for V and (ii) T is linear.) (b) Give a concrete example of vector spaces V and W, a basis {₁,..., Un} of V, and linear transformation T such that {T(v₁),..., T(vn)} is not a basis for range(T).

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a Let W be the sub space of V sp anned by T v 1 T Un Then W range ... View the full answer