4.5 Summary
So, do your eyes hurt after finally using them for the first time? Mine sure did when I first learned about matrices as linear transformations. Here’s a part where the abstract viewpoint pays off for the first time, and trust me, it’ll pay even more dividends later.
Let’s recap this chapter quickly.
We’ve learned that besides a table of numbers, a matrix can represent linear transformations:
where the columns of A describe the images of the basis vectors under the linear transformation x →Ax.
Why on Earth is this useful for us? Think of mathematics as a problem-solving tool. The crux of problem-solving is often finding the proper representation of our objects of interest. Looking at matrices as a way to transform data gives us the much-needed geometric perspective, opening up a whole new avenue of methods.
When looking at matrices this way, we quickly understand why matrix multiplication is defined as it is. The definition
is daunting...
