# Compute the image of the house under the transformation ABC.

http://puu.sh/54hzk.png is the question at hand.

Can someone explain this to me? Is this just knowledge of the matrix transformations such as reflections and shears? What order are the transformations done in?

It depends on what you mean by order. In general, matrices are not commutative, so when you are transforming some vector by $ABC$, then $ABC\vec{v} \neq ACB \vec{v}$ and so on. You should apply this transformation strictly as $ABC\vec{v}$. However, matrices are associative, so once you have the matrix multiplication set up, you can multiply adjacent matrices in pairs. For example, $ABC = (AB)C = A(BC)$.

The matrices do not correspond to the image -- they transform any vector in $\mathbb{R}^2$ into another vector. You can represent the points on the house by vectors, thus allowing you to "apply ABC to the house". For example, I might choose the point on the top right corner of the house, which is the vector $$\left( \begin{array}{c} 1 \\ 1.5 \\ \end{array} \right)$$ To find where that point of the house ends up after applying the transformation, I would multiply: $$\left( \begin{array}{c} x \\ y \\ \end{array} \right) = ABC\left( \begin{array}{c} 1 \\ 1.5 \\ \end{array} \right)$$

Here's a hint/rough sketch of how to apply $C$ that should give you an idea about what to do to complete the problem. My suggestion would be to not multiply the matrices together. Try to describe in words what each matrix does to a vector $v$ in the plane. Each of these matrices does a very specific and easy to describe thing to vectors in the plane. For example, for any real numbers $x$ and $y$

$$\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix}x+y \\ y\end{pmatrix}.$$

So if we think of the vector $(x,y)^T$ as the point $(x,y)$ in the $xy$-plane in the usual manner, then you can visualize $C$ as taking the point $(x,y)$ and moving it horizontally (i.e., in the $x$ direction) by $y$ units. (Note: Here the notation $(x,y)^T$ denotes the transpose of $(x,y)$ for convenience, i.e., $(x,y)^T = \begin{pmatrix}x\\y\end{pmatrix}$.)

Now think about what this does to the points on the house from your picture. If you can't imagine what this looks like yet, that's OK. It will help to plot the points $C(x,y)^T$ in a plane where, say, you first only look at points $(x,y)$ that are the corners of the polygons in your figure (the corners of the house and door, the vertices of the triangle forming the roof, etc.). You can then get what the image of the house should look like by "connecting the dots". Note from what we saw above, the larger the $y$-value of the point $(x,y)$ the farther to the right its image $C(x,y)^T$ will be shifted. So you should expect the image of the house under $C$ to look similar to your current house but leaning to the right.

Once you have that settled, move on to the next matrix $B$ and consider what it does to the house. Again, my suggestion would be to first look at what $B$ does to an arbitrary vector $v = (x,y)^T$ and then apply this information to the problem.

You can do all of this for the matrix $ABC$ after multiplying the three matrices together, too, but it's harder to visualize how $ABC$ transforms vectors than it is to visualize how $A$, $B$, and $C$ do so separately.

Edit: Forgot to respond to your second question (about the order in which you multiply the matrices together). Matrix multiplication is associative but not commutative. In other words, for any $n \times n$ matrices $X$, $Y$, and $Z$ $$X(YZ) = (XY)Z\qquad\qquad\text{(Associativity)},$$ and so you can compute $XYZ$ by first computing $XY$ and then right multiplying by $Z$ or by first computing $YZ$ and then left multiplying by $X$. However, matrix multiplication not being commutative means that sometimes $$XY \neq YX.$$ So you couldn't, for example, compute $ABC$ by computing $AC$ and then right multiplying by $B$, since it's not necessarily true that $ACB = ABC$.