Finding a row vector and calculating the trace of a matrix

[hen93]

Hi there, I've been having trouble with 2 algebra questions, I was hoping someone here could give me a hand.

Homework Statement

(i) Consider the function R2 → R2 defined by f (x, y) = (3x − 4y, x − 2y). Let
v = (a, b) be the vector such that f (v) = (4, 6).
Find the vector v and hence calculate a + b.

(ii) Let f : R3 → R3 be a projection onto the xz-plane. Choose your favourite
basis E for R3 and calculate the matrix A of f with respect to E.
Calculate trace(A)

Homework Equations

N/A.

The Attempt at a Solution

(i) I am positive that this must be incorrect but i took that 3a - 4b =4 and a -2b = 6. Solving for a+b =-15.

(ii)I understand that the trace of a matrix is the sum of all the diagonal entries starting from the top left corner, but the phrasing question has left me clueless.

Any help would be greatly appreciated.
Thank you.

 

[vela]

Hi there, I've been having trouble with 2 algebra questions, I was hoping someone here could give me a hand.

Homework Statement

(i) Consider the function R2 → R2 defined by f (x, y) = (3x − 4y, x − 2y). Let
v = (a, b) be the vector such that f (v) = (4, 6).
Find the vector v and hence calculate a + b.

(ii) Let f : R3 → R3 be a projection onto the xz-plane. Choose your favourite
basis E for R3 and calculate the matrix A of f with respect to E.
Calculate trace(A)

Homework Equations

N/A.

The Attempt at a Solution

(i) I am positive that this must be incorrect but i took that 3a - 4b =4 and a -2b = 6. Solving for a+b =-15.

You did it correctly. Why do you think it's wrong?

(ii)I understand that the trace of a matrix is the sum of all the diagonal entries starting from the top left corner, but the phrasing question has left me clueless.

What specifically is confusing you? Do you know what the difference between a linear transformation and the matrix that represents it is?

 

[hen93]

You did it correctly. Why do you think it's wrong?

What specifically is confusing you? Do you know what the difference between a linear transformation and the matrix that represents it is?

Sorry, I just thought that was to simple to be correct.
I think that I do, just that without any numbers it does not make any sense to me.

 

[vela]

Well, coming up with the numbers is the whole problem. So why not start as suggested and pick your favorite basis for R³. Do you know how to find the matrix once you've chosen a basis? If not, that's what you need to look into.