Linear Transformation

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Linear transformation [itex]T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4[/itex]

Find the standard matrix A for T

[tex]
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)
[/tex]

[tex]
\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}
[/tex]


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.
 

Answers and Replies

  • #2
LCKurtz
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Linear transformation [itex]T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4[/itex]

Find the standard matrix A for T

[tex]
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)
[/tex]

[tex]
\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}
[/tex]


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.
 
  • #3
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That's what I'm really confused about. How do you show the transformation with the matrices?
 
  • #4
LCKurtz
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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
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Okay, thank you.
 
  • #6
Ray Vickson
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Linear transformation [itex]T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4[/itex]

Find the standard matrix A for T

[tex]
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)
[/tex]

[tex]
\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}
[/tex]


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
[tex] 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}
[/tex]

RGV
 

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