Standard Matrix A for Linear Transformation T: R^3 to R^4

[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 sort of convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

Thanks.

 

[LCKurtz]

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 sort of convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

Thanks.

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.

 

[mateomy]

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

 

[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 $T(X) = AX$

 

[mateomy]

Okay, thank you.

 

[Ray Vickson]

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 sort of convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

Thanks.

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