only if it is surjective. Solution T : V ---> W is a linear transformation. We h

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T : V ---> W is a linear transformation.    We have that V/ker T is isomorphic toT(V).----------(1)    Therefore dim(V/ker T) = dim(T(V)).------------------(2)      Also note that T(V) is a subspace ofW. We also have that if W' is a subspace of a vector space Wand dim W' = dim W then W' = W.---------(3) Now it is given that T is injective and hence ker T = 0.    So, V/ker T = V/(0) = V. So (1) will become that V is isomorphic to T(V).       Then by (2), dim V = dimT(V).-----(4)    It is also given that dim V = dim W. Then by (4) dim T(V) = dim W Since T(V) is a subspace of W, by (3) , T(V) =W. This implies that T is surjective

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