Math Help - Linear transformations and their inverse

1. Linear transformations and their inverse

A couple questions:

1)Consider the transformation T from R^2 to R^3 given by T [x1, x2] = x1[1, 2, 3] + x2[4, 5, 6]. Is this transformation linear? If so, find its matrix. Note: these matrices are vertical, I just didn't know how to type them vertically.

2)Find an n x m matrix A such that Ax = 3x, for all x in R^n. Note: x is a vector, I just didn't know how to put a vector arrow above it.

Thanks

2. Originally Posted by noles2188
A couple questions:

1)Consider the transformation T from R^2 to R^3 given by T [x1, x2] = x1[1, 2, 3] + x2[4, 5, 6]. Is this transformation linear? If so, find its matrix. Note: these matrices are vertical, I just didn't know how to type them vertically.
You show that transformation is linear, of course, by showing that the definition of "linear transformation" holds:
Is $T\left(a\begin{bmatrix}x_1 \\ x_2\end{bmatrix}+b\begin{bmatrix}x_3 \\ x_4\end{bmatrix}\right)= aT\left(\begin{bmatrix}x_1 \\ x_2\end{bmatrix}\right)+ bT\left(\begin{bmatrix}x_3\\x_4\end{bmatrix}\right )$?

To write it as a matrix, find $T\left(\begin{bmatrix} 1 \\ 0\end{bmatrix}\right)$ and $T\left(\begin{bmatrix} 0 \\ 1\end{bmatrix}\right)$. That will give you the two columns of the matrix.

2)Find an n x m matrix A such that Ax = 3x, for all x in R^n. Note: x is a vector, I just didn't know how to put a vector arrow above it.
This is easy! Think of it as $A\vec{x}= 3 I\vec{x}$ where I is the identity matrix.

Thanks
To see the LaTex code I used for these, click on each formula.