Finding a Matrix for Successive Transformations

[sara_87]

Homework Statement

find a single matrix that performs the indicated successions of operations:
expands by a factor of 5 in the y-direction, then shears with factor 2
in the y-direction

Homework Equations

The Attempt at a Solution

first for the expansion:
(x,y) maps to (x,5y)
then for the shear:
(x,5y) maps to (x, (2x+5y)

i think its right but I am not quite sure.
thank you.

 

[HallsofIvy]

You have written down the transformations correctly but you haven't answered the question! You were asked to find a matrix. What matrix does that?

What matrix changes (x,y) to (x, 2x+ 5y)? In other words, find the a, b, c,d such that
$$\left[\begin{array}{cc} a & b \\ c & d\end{array}\right]\left[\begin{array}{c} x \\ y\end{array}\right]= \left[\begin{array}{c} x \\ 2x+ 5y\end{array}\right]$$
Multiplying the left side will give you two equations for a, b, c, d but remember they must be true for all x and y. Comparing corresponding coefficients will give you four very simple equations for the a, b, c, d.

 

[sara_87]

ok so multipying the left side gives the matrix:
[(ax+by),(cx+dy)]
so ax+by=x
cx+dy=2x+5y

now what? ;)

 

[Big-T]

You might separate the equations, in order to get:
ax=x, by=0, cx=2x and dy=5y, which ought to be solvable.