Linear Transformation

1. Sep 4, 2012

mateomy

Linear transformation $T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4$

Find the standard matrix A for T

$$T\left(x_1,x_2,x_3\right)\,=\,\left(x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3\right)$$

$$\mathbf{v}\,=\,\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}\,\,\,\,T\left(\mathbf{v}\right)\,=\,T\,\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}\,=\,\begin{bmatrix} x_1 + x_2 + x_3\\ x_2 + x_3\\ 3x_1 + x_2\\ 2x_2 + x_3 \end{bmatrix}\,=\,\begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 3 & 1 & 0\\ 0 & 2 & 1 \end{bmatrix}$$

Is that correct? I feel like I'm just walking down a blind alley with this problem. The text is sorta convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

Thanks.

2. Sep 4, 2012

LCKurtz

Your last = sign has a 4 by 1 matrix equal to a 4 by 3 matrix. You left something out. There doesn't seem to be a question in your post but once you fix that last matrix equality your work looks correct.

3. Sep 4, 2012

mateomy

That's what I'm really confused about. How do you show the transformation with the matrices?

4. Sep 4, 2012

LCKurtz

You just need the matrix $$X = \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$$

column matrix on the right of that last matrix. Then, calling your matrix $A$ you have $T(X) = AX$

5. Sep 4, 2012

mateomy

Okay, thank you.

6. Sep 4, 2012

Ray Vickson

It would have been correct if you had written
$$T(\mathbf{v}) = \begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 3 & 1 & 0\\ 0 & 2 & 1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$$

RGV