# Linear transformation

• Oct 10th 2009, 10:30 AM
Fredrik373
Linear transformation
Hello! I have som problems with this one. maybe someone can help me?

Find the standard matrix for the linear transformation.

T: R^2 --> R^2 dilates a vector by a factor of 3, then reflects that vector about the line y=x, and then projects that vector orthogonally onto the y-axis.

I am supposed to use this theorem while solving :

"Let T: R^n --> R^m be a linear transformation, and suppose that vectors are expressed in column form.
If e1, e2,...,en are the standard vectors in R^n, and if x is any vector in R^n, then T(x) can be expressed as

T(x)=Ax where A = [T(e1) T(e2) ... T(en)]"

=)
• Oct 10th 2009, 03:44 PM
redsoxfan325
Quote:

Originally Posted by Fredrik373
Hello! I have som problems with this one. maybe someone can help me?

Find the standard matrix for the linear transformation.

T: R^2 --> R^2 dilates a vector by a factor of 3, then reflects that vector about the line y=x, and then projects that vector orthogonally onto the y-axis.

I am supposed to use this theorem while solving :

"Let T: R^n --> R^m be a linear transformation, and suppose that vectors are expressed in column form.
If e1, e2,...,en are the standard vectors in R^n, and if x is any vector in R^n, then T(x) can be expressed as

T(x)=Ax where A = [T(e1) T(e2) ... T(en)]"

=)

First it dilates by $3$:

$\left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]$

Then it reflects about $y=x$:

$\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]$

Then it projects orthogonally onto the y-axis:

$\left[\begin{array}{cc}0&0\\0&1\end{array}\right]\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]$

Putting it all together gives us:

$\left[\begin{array}{cc}0&0\\3&0\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]$
• Oct 10th 2009, 09:04 PM
Fredrik373
Thank you very much! =)