Finding the matrix transformation of T

[mr_coffee]

Ookay i don't understand this at all...Sorry i forgot how to use LaTex to form matrices so bare with me...
There is an example in the book that has:
v =
x
y


T[v] =
5x-y
4x - 5/2y

so, they said thus,
T[v] =
5 -1
4 -5/2
*
x
y

so they say:
A =
5 -1
4 -5/2

of T

which makes sense, but in the book first question:
T(v) =
3x -y
2y +x

A =
3 -1
1 2

what the heck? why isn't it
3 -1
2 1

Also there was another one:
T(v) =
y-x
x+y

A =
-1 1
1 1

why isn't it just
A =
1 -1
1 1
?

why did they switch the x and y? and in the example they didn't do anyhting different.

 

[Hurkyl]

Let's look at

$$T(v) = \left( \begin{array}{c}3x - y \\ 2y + x \end{array} \right)$$

in particular. You're trying to find a matrix A such that T(v) = Av.

The book winds up with the matrix:

$$A = \left( \begin{array}{cc}3 & -1 \\ 1 & 2\end{array} \right)$$

while you think it should be

$$A = \left( \begin{array}{cc}3 & -1 \\ 2 & 1\end{array} \right)$$

Right?

Well, have you tried computing Av for each of these matrices? Are either of them equal to T(v)?


Incidentally, do you think that the following S and T are the same linear transformation?

$$T\left( \begin{array}{cc}x \\ y\end{array} \right) = \left( \begin{array}{c}3x - y \\ 2y + x}\end{array} \right)$$

$$S\left( \begin{array}{cc}x \\ y\end{array} \right) = \left( \begin{array}{c}3x - y \\ x + 2y}\end{array} \right)$$

 

[mr_coffee]

thanks for the responce, but I'm so lost, the example in the book just shows how you can find A if you are given: T(i), w, and T(w), i tried to go in reverse and it isn't working out right! Do you know if you can explain to me how you find the Av of each matrix? Or is there somthing online? I searched and couldn't find any good tutorial. I understand how to do these problems which say: assume that T: R^2->R^2 is a linear transformation use the information to determine T[x y]^T for all [x y]^T and find the matrix of T:
http://show.imagehosting.us/show/898600/0/nouser_898/T0_-1_898600.jpg

 

[Hurkyl]

The point of the matrix representation of a linear transformation is (in the R² -> R² case) that:

$$T \left( \begin{array}{c c}x \\ y \end{array} \right) = A \left( \begin{array}{c c}x \\ y \end{array} \right)$$

where T is your linear transformation and A is its matrix representation.


If you write down an arbitrary 2x2 matrix for A, then from:

$$T \left( \begin{array}{c c}x \\ y \end{array} \right) = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{c c}x \\ y \end{array} \right)$$

you can solve for a, b, c, and d if you know T.

Do you know if you can explain to me how you find the Av of each matrix?

I have no idea what this means.

 

[mr_coffee]

ahh i made it way too complicated, thanks for the help!