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Let T: W -> V be a linear transformation and T(w_1) = v_1 for some w_1 \in W and v_1 \in V. Now set S = {w \in W | T(w) = T(w_1) = v_1}. Prove that S = w_1 + Kernel(T) = {w_1 + a | a \in Ker(T)}.

Let w \in S. T(w_1) = T(w) = T(w) + T(a) = T(w_1) + T(a) = T(w_1 + a) = v_1, where a \in Ker(T).

Thus

S_1 = {w \in W | T(w) = T(w_1) = v_1}; S_2 = {w_1 + a | a \in Ker(t)}

S_1 = S_2 = S and we're done. Correct?

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# Homework Help: Linera Transformation Questions

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