2. (a) Let V be the subspace of Mat22 (R) spanned by 1 0 -20 -4...

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2. (a) Let V be the subspace of Mat22 (R) spanned by 1 0 -20 -4 {(+49) (4) (13) (439)}. 2 -1 2 (i) Find a basis for V consisting of matrices from those listed above. What is the dimension of V? (ii) Let W be the set of all 2 2 matrices of the form [6] 2x x x R. X x Show that W is a subset of V. [4] (iii) Show that W is a subspace of V. Begin your answer by stating what condition(s) you have to verify. [5] (b) Let U and V be vector spaces and let v be a non-zero vector of V. Is there a linear transformation UV such that the set {uU (u) = v} == is a subspace of U? Justify your answer. [5] 2. (a) Let V be the subspace of Mat22 (R) spanned by 1 0 -20 -4 {(+49) (4) (13) (439)}. 2 -1 2 (i) Find a basis for V consisting of matrices from those listed above. What is the dimension of V? (ii) Let W be the set of all 2 2 matrices of the form [6] 2x x x R. X x Show that W is a subset of V. [4] (iii) Show that W is a subspace of V. Begin your answer by stating what condition(s) you have to verify. [5] (b) Let U and V be vector spaces and let v be a non-zero vector of V. Is there a linear transformation UV such that the set {uU (u) = v} == is a subspace of U? Justify your answer. [5]