1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving that a matrix is an inverse map

  1. Jun 29, 2013 #1
    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.

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

    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).
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?