We have a linear transformation T : M 2 &#x00D7;<!-- × -->

Davon Irwin

Davon Irwin

Answered question

2022-06-21

We have a linear transformation T : M 2 × 2 ( F ) F by T ( A ) = t r ( A ).
We want to compute the matrix representation [ T ] from α to γ coordinates.
M 2 × 2 has the standard ordered basis " α" for a 2 × 2 matrix. F has the standard basis " γ" for a scalar.
My understanding is that any matrix A in the space M 2 × 2 can be represented by the linear combination:
a α 1 + b α 2 + c α 3 + d α 4
and the t r ( A ) can be written as:
( a + c ) γ .
I'm not sure how to get the matrix representation from this.

Answer & Explanation

podesect

podesect

Beginner2022-06-22Added 20 answers

Our static map is
T : M 2 × 2 ( F )  F   A  tr  ( A ) 
The basis α for M 2 × 2 ( F ) is
α = { [ 1 0 0 0 ] , [ 0 1 0 0 ] , [ 0 0 1 0 ] [ 0 0 0 1 ] } 
and the basis γ for F is γ = { 1 }.
Now observe that
T ( [ 1 0 0 0 ] ) = 1 T ( [ 0 1 0 0 ] ) = 0 T ( [ 0 0 1 0 ] ) = 0 T ( [ 0 0 0 1 ] ) = 1 
This implies that [ T ] α γ is the 1 × 4 matrix
[ T ] α γ = [ 1 0 0 1 ] 
We may understand trace as matrix multiplication because of this. Recall that
A = [ a b c d ] = a [ 1 0 0 0 ] + b [ 0 1 0 0 ] + c [ 0 0 1 0 ] + d [ 0 0 0 1 ] 
which means that relative to α the matrix A is observable as the vector
[ a b c d ] 
So
tr  ( A ) = [ 1 0 0 1 ] [ a b c d ] = a + d

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