My twofold hope then is, that the creating you do in meeting the challenges of this course will help you to reach greater understanding of the material. And two, that, that greater understanding will give you the ability to create even more.
In addition, I want to highlight three specific concepts from programming that
are relevant to this course. Data abstraction, procedure abstraction, and what I would call procedural sophistication.
In linear algebra, data abstraction gives us the concept of a field, think of a field as a class that implements the operations plus, minus, times, and divide.
As long as a class's implementation of these operations satisfies certain algebraic rules, we can use it to do linear algebra. (new added)
And the field of complex numbers comes up in signal processing and in analyzing
dynamic processes, processes that unfold over time.(How to understand?)
There's a lot to learn here, but it boils down to learning certain kinds of addition and multiplication operations.
**In this part of the course, in order to better learn these operations, you'll implement your own vectors and matrices. **Since you'll be working from a template, all you have to do is fill in the implementations of these operations.
In the middle third of this course, there are a lot of concepts rolling around, coordinate representation, bases, dimension, rank. Each concept on its own is not so hard, but it might be hard to keep them straight, to develop an understanding of the relationships between these concepts.
Third, if you've done a fair bit of programming, I expect you've developed what I call, procedural sophistication.
In the middle third of the course, while we're dealing with all these concepts
swirling around, you're going to see some fairly complicated mathematical proofs. Understanding these proofs, might require you to think harder than your used to,
but it turns out that many of these mathematical proofs are procedures in disguise.
In the final third of the course, we'll start opening up those black boxes and learning the procedures that are inside.
The first third of the course consists of a lot of mechanics and definitions, working with vectors and matrices.
The middle third of the course, the concepts we will study will illuminate the topic enabling us to answer many of the questions that have come up so far.
The real solutions to the computational problems come in the final third of the
course, where we study what makes the black box, the solver work.
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