1) Determine if the set U is a subspace of the vector V. If yes show the 3 properties, if no then say why.

a) Let V be the vector space of all continuously differentiable functions on the real line. U is the subset of V such that U = {f : f ' = 3f}.

b) Let V be the vector space of all continuously differentiable functions on the real line. U is the subset of V such that U = {f : f ' = f+1}.

c) V=M2,2. U is the set of 2x2 invertible matrices.

d) V=M2,2. U is the set of 2x2 symmetric matrices.

2) Let T:V->W be a linear transformation. Prove that T is one-to-one if and only if the rank of T equals the dimension of V.

a) Let V be the vector space of all continuously differentiable functions on the real line. U is the subset of V such that U = {f : f ' = 3f}.

b) Let V be the vector space of all continuously differentiable functions on the real line. U is the subset of V such that U = {f : f ' = f+1}.

c) V=M2,2. U is the set of 2x2 invertible matrices.

d) V=M2,2. U is the set of 2x2 symmetric matrices.

2) Let T:V->W be a linear transformation. Prove that T is one-to-one if and only if the rank of T equals the dimension of V.

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