# Transformation

• Jun 18th 2008, 10:01 AM
Simplicity
Transformation
If $\displaystyle T: \begin{pmatrix} x \\y \\z \end{pmatrix} ->\begin{pmatrix} x \\x+y \\x+z \end{pmatrix}$ then what does $\displaystyle T \begin{pmatrix} x+y \\y \\y+z \end{pmatrix}$ equal?

Can you show in steps as I find this topic confusing. Thanks in advance.
• Jun 18th 2008, 10:10 AM
bobak
Still breathing after this morning huh ?

$\displaystyle T \begin{pmatrix} x+y \\y \\y+z \end{pmatrix} = \begin{pmatrix} x+y \\x + 2y \\x +2y+z \end{pmatrix}$

Bobak
• Jun 18th 2008, 10:47 AM
Moo
Hello !

Quote:

Originally Posted by Air
If $\displaystyle T: \begin{pmatrix} x \\y \\z \end{pmatrix} ->\begin{pmatrix} x \\x+y \\x+z \end{pmatrix}$ then what does $\displaystyle T \begin{pmatrix} x+y \\y \\y+z \end{pmatrix}$ equal?

Can you show in steps as I find this topic confusing. Thanks in advance.

I'd suggest you to rewrite that (Wink) because using the same x & y & z stuff makes things complicated !

$\displaystyle T: \begin{pmatrix} m \\ n \\ p \end{pmatrix} \mapsto \begin{pmatrix} m \\ m+n \\ m+p \end{pmatrix}$

$\displaystyle T(\textbf{v})=T \begin{pmatrix} x+y \\ y \\ y+z \end{pmatrix}$

$\displaystyle x+y=m$
$\displaystyle y=n$
$\displaystyle y+z=p$

---> $\displaystyle m=x+y$
$\displaystyle m+n=(x+y)+y=x+2y$
$\displaystyle m+p=(x+y)+(y+z)=x+2y+z$

Got it ? =)
• Jun 18th 2008, 12:11 PM
Simplicity
Quote:

Originally Posted by Moo
$\displaystyle y=n$

How did you get $\displaystyle y=n$?
• Jun 18th 2008, 12:13 PM
Moo
Quote:

Originally Posted by Air
How did you get $\displaystyle y=n$?

Ow ^^

$\displaystyle T: \begin{pmatrix} {\color{red}m} \\ {\color{blue}n} \\ {\color{green}p} \end{pmatrix} \mapsto \begin{pmatrix} m \\ m+n \\ m+p \end{pmatrix}$

$\displaystyle T(\textbf{v})=T \begin{pmatrix} {\color{red}x+y} \\ {\color{blue}y} \\ {\color{green}y+z} \end{pmatrix}$
• Jun 18th 2008, 12:46 PM
bobak
Can you find the 3 by 3 matrix that represents the transformation ?

Bobak
• Jun 18th 2008, 01:11 PM
Simplicity
Quote:

Originally Posted by bobak
Can you find the 3 by 3 matrix that represents the transformation ?

Bobak

(Worried)
• Jun 18th 2008, 01:37 PM
bobak
As $\displaystyle T$ is a liner transformations there exists and matrix such that $\displaystyle A \textbf{v} = T( \textbf{v} )$ to find this matrix you need to find the images of the vectors $\displaystyle i, j$ and $\displaystyle k$ under $\displaystyle T$. This all lies down the understanding how matrix multiplication works.

I do this row by row, you see that the $\displaystyle i$ vector is preserved under the transformation, so the first row is one $\displaystyle i$ combined linearly with no j's or k's

so the first row is $\displaystyle \left(\begin{array}{ccc}1 & 0 & 0 \\? & ? & ? \\? & ? & ?\end{array}\right)\left(\begin{array}{c}x \\y \\z\end{array}\right)$

for the next row you combining one $\displaystyle i$ with one $\displaystyle j$ and no k's

so we have $\displaystyle \left(\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\? & ? & ?\end{array}\right)\left(\begin{array}{c}x \\y \\z\end{array}\right)$

Then similarly for the next row your combining 1 i with 1 k.

giving $\displaystyle \left(\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right)\left(\begin{array}{c}x \\y \\z\end{array}\right)$

to check this is correct just multiple the two.

$\displaystyle \left(\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right)\left(\begin{array}{c}x \\y \\z\end{array}\right) = \begin{pmatrix} x \\x+y \\x+z \end{pmatrix}$

so $\displaystyle T \begin{pmatrix} x+y \\y \\y+z \end{pmatrix}$ is given by $\displaystyle \left(\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right) \begin{pmatrix} x+y \\y \\y+z \end{pmatrix}$

You following?

Bobak
• Jun 18th 2008, 01:44 PM
Simplicity
Quote:

Originally Posted by bobak
You following?

Bobak

Yes! Much better than the book too. (Clapping)

Thanks!