show that the is one to one correspondence between the space L(V,W) and Mmxn or

Question

show that the is one to one correspondence between the space L(V,W) and Mmxn
or show that there is an ismorphim between the set of linear maps and the set of mxn matrices

Explanation / Answer

Assuming V = R^m , W = R^n Consider any basis of V = v1, .. vm Consider any basis of W = w1, .. .wn Consider any linear map T: V -> W Consider the vectors T(v1),T(v2)...T(vm) When we specify these we specify the whole map as T is linear To specify each of these we require n reals a1 , a2, .. an as T(v1) = a1w1 + a2w2 + .. anwn similarly T(v2) = b1w1 + b2w2 + .. bnwn ... ... So to specify T we need mxn scalars numbers these can be arranged in a matrix and the multiplication will describe the transform like A = matrix of mxn A (v1,v2..vm)transpose = (T(v1),...) All these relations are bijectives so given a transform we can find A by the similar computations Hence proved

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