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Math · Advanced Math
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Let V and W be vector spaces over R with dim(V) = n and dim(W) = m, and let Lin(V,W) denote the set of all linear transformations L : V → W We can uniquely express any vEV in terms of the basis B as a linear combination The coordinate representation of v with respect to B is the column vector BV = Recall the following theorem from class: Theorem 0.1. Let V and W be vector spaces with bases B = {u! C-w,..., wmi, respectively. Any linear map L:V-- W can be expressed as , י un) an where cLB is the matrix representation of L with respect to bases B and C, iven bu the m × n-matrix Show that the map F : Lin(V, W) → Rmxn defined by is a linear isomorphism. For the basis elements δǐj defined above, what is FGj)?

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