Finding the Standard Matrix A of a Linear Transformation T

[x.x586]

Homework Statement

Let T be a linear transformation from R3 to R3. Suppose T transforms (1,1,0) ,(1,0,1) and (0,1,1) to (1,1,1) (0,1,3) and (3,4,0) respectively.

Find the standard matrix of T and determine whether T is one to one and if T is onto

Homework Equations

The Attempt at a Solution

taking a 3x3 matrix entries [x1,x2,x3;x4,x5,x6] and multiply that by a 3x3 matrix with entries [1,1,0;1,0,1;0,1,1] and set that equal to a matrix with entries [1,0,3;1,1,4;1,3,0] and then got a system of equations from there by multiplying the left side out. And then set up an augmented matrix and used row reduction to find corresponding entries for A?

 

[jfgobin]

If you call:

<br /> V_1= \begin{pmatrix} 1\\1\\0\end{pmatrix},<br /> V_2= \begin{pmatrix} 1\\0\\1\end{pmatrix},<br /> V_3= \begin{pmatrix} 0\\1\\1\end{pmatrix}\\<br /> B_1=\begin{pmatrix} 1\\1\\1\end{pmatrix},<br /> B_2=\begin{pmatrix} 0\\1\\3\end{pmatrix},<br /> B_3=\begin{pmatrix} 3\\4\\0\end{pmatrix}<br />
​

Then you may write that

<br /> T\cdot\left ( V_1 V_2 V_3\right) = \left ( B_1 B_2 B_3 \right )<br />
​

Observe a few things about \left ( V_1 V_2 V_3\right) and you should be on your way to finding the solution.