linear algebra question

Hi, Suppose T is a linear map from R^2 to R^3 and it satisfies T(1,1) = (1,1,0) and T(2,3) = ( -1,2,1). What is the matrix T and the formula for T(x,y)?

My troubl with this question is that I cant seem to find a 3 by 2 matrix such that when multiplied by (x,y) gives both vectors in R^3 as above

Quote:

Originally Posted by ferico

Hi, Suppose T is a linear map from R^2 to R^3 and it satisfies T(1,1) = (1,1,0) and T(2,3) = ( -1,2,1). What is the matrix T and the formula for T(x,y)?

My troubl with this question is that I cant seem to find a 3 by 2 matrix such that when multiplied by (x,y) gives both vectors in R^3 as above

Let $T = \left( \begin{array}{cc} a & b \\ c & d \\ e & f \end{array} \right)$

then, $T {1 \choose 1} = \left( \begin{array}{c} a + b \\ c + d \\ e + f \end{array} \right) = \left( \begin{array}{c} 1 \\ 1 \\ 0\end{array} \right)$

and

$T {2 \choose 3} = \left( \begin{array}{c} 2a + 3b \\ 2c + 3d \\ 2e + 3f \end{array} \right) = \left( \begin{array}{c} -1 \\ 2 \\ 1 \end{array} \right)$

equating components, we get the systems:

$a + b = 1$ .............(1a)
$2a + 3b = -1$ .........(2a)

$c + d = 1$ ..............(1b)
$2c + 3d = 2$ ............(2b)

$e + f = 0$ ..............(1c)
$2e + 3f = 1$ ...........(2c)

these are each simultaneous equations that you should be able to solve

But this is precisely the question. I don't know which a and b to choose such that a+b =1 and 2a +3b = -1.

nvm lol i got it "simultaneous equations" i can solve it. Thanks alot for your help