Linear Map w/ Matrix: Solve for a + d

[KevinL]

Homework Statement

Consider the map L from the space of 2x2-matrices to R given by:

L([a b]) = a+ d
([c d])

For clarity, that's L(2x2 matrix) = a + d

The Attempt at a Solution

Im confused how any function of a matrix could possibly equal addition of two scalars, and thus have no idea where to begin.

 

[Mark44]

What's the entire problem? Your problem statement is incomplete.

 

[Mark44]

To answer your question, this transformation takes a 2x2 matrix as input, and produces a single number as output. It takes the entries in the upper left and lower right corner and produces their sum as its output.

 

[KevinL]

Sorry not sure how I missed that. The question is "Is L a linear map?"

I think I MAY have it now. I need to show linearity by proving L(a+b+c+d)=L(a) + L(b) + L(c) + L(d)

So, L(a+b+c+d) = a 2x2 matrix where each corner is (a + b + c + d)

L(a) is a 2x2 matrix w/ each corner containing a, L(b) is a 2x2 matrix w/ each corner containing b, and same thing for L(c) and L(d). These two things do in fact equal each other, so its a linear map.

 

[Mark44]

I agree that L is linear, but not the way you did it.

What you wrote "So, L(a+b+c+d) = a 2x2 matrix where each corner is (a + b + c + d)
" doesn't make any sense. a, b, c, and d are real numbers, the entries in a 2x2 matrix. a + b + c + d is a single real number. The domain of L is not real numbers.
So L(a + b + c + d) doesn't make any sense, nor is L(whatever) = a 2x2 matrix. L(whatever) is a number.

Also, how can each corner be (a + b + c + d)?

What you want to do is something like this:
Let A and B be 2x2 matrices.
Now show that L(A + B) = L(A) + L(B).
The use of cap letters for matrices prevents confusion with a and b that represent entries in a matrix.