Proving that a matrix is an inverse map

[Emspak]

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:

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

Homework Statement

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

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

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).

 

[lippyka]

The best way to approach this problem is to break it down into smaller pieces. First, you need to understand what the matrix representation map does and how it works. This is a map that assigns a matrix to a linear map, with the entries in the matrix corresponding to the coefficients of the linear map in terms of the given bases. To find the inverse, you need to find the linear map that corresponds to a given matrix. To do this, start by writing down the definition of the linear map L_A. You are given that if (x_1,x_2,...,x_n)∈K, (after identifying this row matrix with the column matrix X) L_A((x_1,x_2,…x_n))=L_A(X)=AX. This means that the linear map L_A takes in a vector X and multiplies it by the matrix A to give a new vector.Now, consider the matrix representation map []ββ′. This is a map from Hom_K(V,W)→M_{n×n}(K), which assigns a matrix to a linear map with entries corresponding to the coefficients of the linear map in terms of the given bases. From this, it should be clear that the inverse of the matrix representation map should be a map from M_{n×n}(K) to Hom_K(V,W), which takes in a matrix A and returns the linear map L_A that corresponds to it. That is exactly what the map L:M_{n×n}(K)→Hom_k(V,V) is doing, so the map L is the inverse of the matrix representation map.