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In: Advanced Math

True or False? If true, give a brief reason why; if false, give a counterexample. (Assume...

True or False? If true, give a brief reason why; if false, give a counterexample. (Assume in all that V and W are vector spaces.)

a. If T : V → W is a linear transformation, then T(0) = 0.

b. Let V be a vector space with basis α = {v1, v2, . . . , vn}. Let S : V → W and T : V → W be linear transformations, and suppose for all vi ∈ α, S(vi) = T(vi). Then S = T, that is, for all v ∈ V , S(v) = T(v).

c. Every linear transformation from R3 to R3 has an inverse. That is, if T : R3 → R3 is a linear transformation, then there exists a linear transformation S : R3 → R3 such that S(T(v)) = T(S(v)) = v for all v ∈ R3 .

d. If T : Rn → Rm is a linear transformation and n > m, then Ker(T) 6= {0}.

e. If T : V → W is a linear transformation, and {{v1, v2, . . . , vn} is a basis of V , then {{T(v1), T(v2), . . . , T(vn)} is a basis of W.

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