Proving that a matrix is an inverse map

  • Thread starter Emspak
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OK, this is one where I am having trouble starting because I am not sure I am reading the question correctly to begin with. So start with:

[itex]M_{m×n}(K)[/itex] denotes the set of all matrices with m rows and n columns with entries in the field K. Let β⊂V and β′⊂W be bases of vector spaces V and W respectively

Let [itex]dim_KV=n[/itex] and [itex]dim_KW=m[/itex]. If [itex]f:W→W[/itex] is a linear map then let [itex][]ββ′:Hom_K(V,W)→M_{n×n}(K)[/itex] denote the matrix representation map.

Homework Statement



Let [itex]A∈M_{n×n}(K)[/itex] and let [itex]L_A:K^n→K^n[/itex] be a linear map such that if [itex](x_1,x_2,…,x_n)∈K[/itex], (after identifying this row matrix with the column matrix X) [itex]L_A((x_1,x_2,…x_n))=L_A(X)=AX[/itex].

Show that the map [itex]L:M_{n×n}(K)→Hom_k(V,V)[/itex] such that [itex]L(A)=LA[/itex] is the inverse map of the matrix representation map [itex][]ββ′ [/itex] where [itex]β=\{e_1,e_2,…,e_n\}[/itex] is the canonical basis of [itex]K_n[/itex].

OK, this is where I am not sure if I am even reading this question right. It's asking to show that the linear map L(A)=LA is the inverse of the matrix representation map. So that means I have to take a linear map of any matrix, A, but I guess I need to find the matrix representation of the matrix? This makes no sense whatsoever to me right now, and it's getting more and more frustrating by the minute.

I understand -- I think - -finding a matrix representation. But this I can't even figure out where to start. Maybe there's a similar question answered on here and that's fine, but I wouldn't even know how to search for the thing. (Picture someone smashing his head against the table right now. I'm sorry to vent).
 

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