# Transformation matrix

• Mar 13th 2011, 09:55 AM
yorkey
Transformation matrix
Hi

I've got this math problem in an past IGCSE exam.. Even though it's question (c), it doesn't rely on any other information above it, so thats the whole question.

What I want to know is, how can I tell what type of transformation will it be just by looking at the matrix?

Thanks
• Mar 13th 2011, 10:28 AM
Soroban
Hello, yorkey!

I don't think you can "eyeball" the matrix.
You might have to do a teensy bit of math . . .

Quote:

$\displaystyle \text{(c) The matrix }\,\begin{pmatrix}0 & 1 \\ \text{-}1 & 0 \end{pmatrix}\,\text{ represents a single transformation.}$

$\displaystyle \text{(1) Describe fully this transformation.}$

Let $\displaystyle (a,b)$ be any vector.

Then: .$\displaystyle (a,b)\begin{pmatrix}0 & 1 \\ \text{-}1 & 0\end{pmatrix} \;=\;(\text{-}b,a)$

It transforms point $\displaystyle P(a,b)$ to the point $\displaystyle Q(\text{-}b,a).$

Code:

          Q      |           *      |           :*    |           : *    |          P           :  *  |          *           a:  *  |        *  :           :    * |    *    :b           :    *|  *        :       ----+------*-----------+----               -b  |      a                   |

Point $\displaystyle \,P$ is rotated $\displaystyle 90^o$ counterclockwise about the Origin.

Quote:

$\displaystyle \text{(ii) Find the coordinates of the image of the point }(5,3).$

. . $\displaystyle (5,3)\begin{pmatrix}0 & 1 \\ \text{-}1 & 0\end{pmatrix} \;=\;(\text{-}3,5)$

• Mar 13th 2011, 10:38 AM
yorkey
Quote:

Originally Posted by Soroban

Let $\displaystyle (a,b)$ be any vector.

Then: .$\displaystyle (a,b)\begin{pmatrix}0 & 1 \\ \text{-}1 & 0\end{pmatrix} \;=\;(\text{-}b,a)$

It transforms point $\displaystyle P(a,b)$ to the point $\displaystyle Q(\text{-}b,a).$

Thank you! I understand all the steps except this one: $\displaystyle (a,b)\begin{pmatrix}0 & 1 \\ \text{-}1 & 0\end{pmatrix} \;=\;(\text{-}b,a)$
Did you replace the variables a and b with any numbers? Sorry if I'm confusing you. I know how to multiply matrices, but how did you do it with variables?