Suppose T: VW is a linear transformation, let S be a subset of V, and let...

 

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Suppose T: VW is a linear transformation, let S be a subset of V, and let T(S) = {T(s) : s S). (a) If T(S) is always linearly independent whenever S is linearly independent, show that T is one-to-one. (b) If T(S) always spans W whenever S spans V, show that T is onto. (e) If T(S) is always a basis of W whenever S is a basis of V, show that T is an isomorphism. Suppose T: VW is a linear transformation, let S be a subset of V, and let T(S) = {T(s) : s S). (a) If T(S) is always linearly independent whenever S is linearly independent, show that T is one-to-one. (b) If T(S) always spans W whenever S spans V, show that T is onto. (e) If T(S) is always a basis of W whenever S is a basis of V, show that T is an isomorphism.