a Prove that for every n n matrix B there exists a linear tr

(a) Prove that for every n n matrix B, there exists a linear transformation T : V --> V such that

(b) Let L(V, V ) be the vector space consisting of all linear transformations from V to V . Use part (a) to prove that L(V, V ) is isomorphic to Rⁿⁿ.

Let V be a vector space and beta -matrix of T and prove that the transformation is unique. (b) Let L(V, V ) be the vector space consisting of all linear transformations from V to V . Use part (a) to prove that L(V, V ) is isomorphic to Rn½n. beta = (f1,...,fn) be a basis of V. (a) Prove that for every n ½ n matrix B, there exists a linear transformation T : V --> V such that B is the

Solution

Beta = (f1,f2,...fn) is a basis of V

Consider any nxn matrix B with {bij}

f1 = b1b11+b2b13+b3b13+...

...

Thus f1, f2 ... can be represented as a linear tranformation uniquely.

Thus there exist a T: V to V such that B is the Beta matrix of T and the transformation is unique.

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b)Consider set of all linear transformations

L(V,V)

Let L(V,V) be mapped to the Beta matrix nxx

Then the mapping is isomorphic

Hence L(V,V) is ismorphic to Rnxn