# Proving that a matrix is an inverse map

1. Jun 29, 2013

### 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 $[]ββ′om_K(V,W)→M_{n×n}(K)$ denote the matrix representation map.

1. The problem statement, all variables and given/known data

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).