Question

Consider linear transformation T. Given that we have:
T([1,1,0)] = [2,5,1,3]
T([1,2,0)] = [-1,1,2,2] and
T([1,2,1)] = [1,0,0,2]
Find the standard matrix of T and use it to determine T([1,1,1]) ... Any ideas would be greatly appreciated :) thanks!

Answer 1

This might help (see attached). This is drawn from the link below

http://www.sinclair.edu/centers/mathlab/pub/findyourcourse/worksheets/215,216/MatrixAndLinearTransformations.pdf

Answer 2

T is a 4X3 matrix, here is example for first transformation
\[\left[\begin{matrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} &c_{3}\\ a_{4}&b_{4} &c_{4}\end{matrix}\right]*\left[\begin{matrix}1 \\ 1\\0\end{matrix}\right] = \left[\begin{matrix}2 \\5 \\ 1 \\ 3\end{matrix}\right]\]
then set up a system of equations for each tranformation
\[a_{1} + b_{1} =2\]
\[a_{1}+2b{1} = -1\]
\[a_{1}+2b_{1} +c_{1} = 1\]
Now you can solve for for a1,b1,c1
do the same thing to find every element in matrix T

Answer 3

Great, thanks guys! dumbcow (I feel like I'm being rude calling you that haha), can I, instead of just using \[\left(\begin{matrix}1\\1 \\ 0\end{matrix}\right)\] and doing it 3 times, can I set it out (in my working - it's for an assignment) as the first matrix you have drawn, multiplied by\[\left[\begin{matrix}1 & 1&1 \\ 1 & 2&2\\0&0&1\end{matrix}\right]\]
equal to \[\left[\begin{matrix}2 & -1&1 \\ 5 & 1&0\\1&2&0\\3&2&2\end{matrix}\right]\] ... or does that not make sense...? :/

Answer 4

no that makes sense, should've thought of that.....either way you will get the same equations and the solving should be fairly straightforward using substitution

Answer 5

great! thanks heaps :D