# Linear Transformation

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.

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LCKurtz
Homework Helper
Gold Member
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.
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.

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

LCKurtz
Homework Helper
Gold Member
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 $T(X) = AX$

Okay, thank you.

Ray Vickson
Homework Helper
Dearly Missed
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.
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