How to Calculate Transition Matrices for Symmetric 2x2 Matrices?

[schmiggy]

Homework Statement

Let V be the vector space of all symmetric 2x2 matrices, and consider the bases.
S = {
[1 0] [0 1] [0 0]
[0 0],[1 0],[0 1]}

B = {
[1 1] [-1 1] [1 0]
[1 2],[ 1 1],[0 1]}
of V.
Find the transition matrix Ps,b. Use your answer to calculate Pb,s.

Homework Equations

a = $\alpha$$_{1}$b$_{1}$ + $\alpha$$_{2}$b$_{2}$ + $\alpha$$_{3}$b$_{3}$ +... + $\alpha$$_{k}$b$_{k}$

The Attempt at a Solution

I honestly don't know where to start. All previous questions like this we've dealt with vectors and not 2x2 matrices..

ie B = (1,3),(2,1)

(1,3) = 1(1,0) + 3(0,1) and (2,1) = 2(1,0) + 1(0,1)

So Ps,b = [1 2] and Pb,s is just the inverse of Ps,b = -1/5[1 -2]
[3 1] [-3 1]

But I don't know how to even start when I'm given 2x2 matrices..

 

[kudrmj]

haha I am doing the same linear assignment. Thought id be awesome and let you know that Ps,b is {(1, -1, 1), (1, 1, 0), (2, 1, 1)}. make an identity matrix to solve the inverse to get Pb,s.

 

[schmiggy]

Haha yeah, I ended up figuring it out - I never usually get a response from this website, I don't know why, yours is the first one I've gotten so thanks.

Let me know if you need a hand with either of the other questions.

 

[kudrmj]

No worries mate, and actually I am pretty good for the other two, was just a bit unsure of Q1 b). did you manage to sus that?

 

[schmiggy]

For 1b) I got
[ 1 2 -1] [-1] = [0]
[-1 -1 1] [ 3] [3]
[-1 -3 2] [ 5] [2]

That's written out pretty garbagety, but hopefully you can decipher it.. it's Pb,s multiplied by (-1,3,5)