Linear Transformations

• Feb 5th 2010, 12:11 AM
temaire
Linear Transformations
Let u = (1,2), v = (3,1) and T: $\displaystyle R^{2}\rightarrow R$ be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)

I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?
• Feb 5th 2010, 02:38 AM
math2009
$\displaystyle S=[\vec{u},\vec{v}],\begin{bmatrix}-3\\4\end{bmatrix}=S(S^{-1}\begin{bmatrix}-3\\4\end{bmatrix})=[\vec{u},\vec{v}]\begin{bmatrix}3\\-2\end{bmatrix}= 3\vec{u} -2\vec{v}$

$\displaystyle T(\begin{bmatrix}-3\\4\end{bmatrix})=T(3\vec{u} -2\vec{v})=3T(\vec{u})-2T(\vec{v})=3\times 4-2\times 5=2$
• Feb 5th 2010, 04:01 AM
HallsofIvy
Quote:

Originally Posted by temaire
Let u = (1,2), v = (3,1) and T: $\displaystyle R^{2}\rightarrow R$ be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)

I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?

Once you have (-3, 4)= au+ bv, T((-3,4))= aT(u)+ bT(v)= 4a+ 5b.