# Linear transformation

• Jul 27th 2009, 12:30 PM
justin016
Linear transformation
Consider the basis S={ v1,v2 } where v1= (-2,1), v2=(1,3) and let
T : R^2 --> R^3 be the linear transformation such that

T(v1) = ( -1,2,0) and T(v2) = (0,-3,5)

Find a formula for T( x1, x2)

Here what I did

let x = (x1,x2) and

c1v1+c2v2=x so

-2c1 +1c2 = x1
1c1 + 3c2 = x2 so i have to express c1 and c2 in term of x1 and x2

i get this system

-2 1 = x1
1 3 = x2
I use ref(a) to get this

1 -1/2 = x1
0 1 = x2
so c2= x2
c1= x1+1/2x2

(x1 +1/2x2)(-1,2,0) + (x2)(0,-3,5) = (-x1-1/2x2, 2x1-2x2, 5x2)

the answer is 1/7(3x1-x2, -9x1-4x2, 5x1+10x2)

what did i do wrong here? thanks in advanced
• Jul 27th 2009, 03:47 PM
Danneedshelp
Hey there,

You have the right idea, you just messed up on your algebra. Try redoing your first system of equations.

Working down to RREF I got...

$\displaystyle \begin{bmatrix}1&0\\0&1\end{bmatrix}$ $\displaystyle =\begin{bmatrix}\frac{x_{2}}{7}-\frac{3x_{1}}{7}\\\frac{2x_{2}}{7}+\frac{x_{1}}{7} \end{bmatrix}$

$\displaystyle T(\vec{x_{1}},\vec{x_{2}})=c_{1}(-1,2,0)+c_{2}(0,-3,5)=$ your books answer!