Linear algebra fun never stops

[stunner5000pt]

In each case find a rotation or reflection which equals the given transformation

Rotation through pi followed by reflection in the X axis
Now is there a way to actually work this out?

The text says the answer is simply reflection on the Y axis.
I tried to visualize by drawing a vector, rotating it pi and then reflecting it on the X axis but it doesn't seem to make sense...

A rotation thru pi would yield a matrix like this
\left(\begin{array}{cc} -1&0 \\ 0&-1 \end{array}\right)How would one incorporate the reflection on the X Axis.

For that matter, how would one incorporate a Y axis reflection or a line y = mx reflection onto a matrix?

 

[Gale]

if you reflect over the x axis, it only effects the y's. specifically it makes them negative. so that matrix would be
\left(\begin{array}{cc} 1&0 \\ 0&-1 \end{array}\right)

when you multiply the two matrices, you get:
\left(\begin{array}{cc} -1&0 \\ 0&-1 \end{array}\right) \left(\begin{array}{cc} 1&0 \\ 0&-1 \end{array}\right) = \left(\begin{array}{cc} -1&0 \\ 0&1 \end{array}\right)

when you reflect over the y axis, then you're just making all the x's negative. which, is the matrix we got as our answer.

 

[0rthodontist]

Here is a neat trick. In any linear transformation, you only need to know where the standard unit vectors go. Let's say you have a linear transformation, and you know from the definition of this transformation that the vector
\left(\begin{array}{cc} 1 \\ 0 \end{array}\right)
should go to
\left(\begin{array}{cc} 0.5 \\ 0.2 \end{array}\right)

Now you have a matrix
\left(\begin{array}{cc} a&b \\ c&d \end{array}\right)
Now what could a and c possibly be so that (1, 0) gets mapped to (0.5, 0.2)? Think about how to multiply a vector by a matrix.

 

[HallsofIvy]

Orthodontist gave you exactly what you need to know. (1, 0) is rotated 180 degrees counter clockwise so it ends up where? Reflecting that in the x-axis changes it to? That's your first column.
(0,1) is rotated 180 degrees clockwise so it ends up where? Reflecting that in the x-axis changes it to? That's your second column.

 

[stunner5000pt]

Here is a neat trick. In any linear transformation, you only need to know where the standard unit vectors go. Let's say you have a linear transformation, and you know from the definition of this transformation that the vector
\left(\begin{array}{cc} 1 \\ 0 \end{array}\right)
should go to
\left(\begin{array}{cc} 0.5 \\ 0.2 \end{array}\right)

Now you have a matrix
\left(\begin{array}{cc} a&b \\ c&d \end{array}\right)
Now what could a and c possibly be so that (1, 0) gets mapped to (0.5, 0.2)? Think about how to multiply a vector by a matrix.

so then i would have to multiply

\left(\begin{array}{cc} 1 \\ 0 \end{array}\right)^T \left(\begin{array}{cc} a&b \\ c&d \end{array}\right) = \left(\begin{array}{cc} 0.5 \\ 0.2 \end{array}\right)

right?

so what if (1,0) was rotated pi/2? I know it would be (0,1)... but ist here a systematic way of doing it? What about pi/4?