The whole course will be about linear combinations of vectors. We will look at progressively more sophisticated ways at looking at them. It is important to note that a lot of our methods will be applicable beyond vectors. In fact Fourier analysis in EE 315 happens to be a use of these approaches for function spaces rather than vector spaces.
In the text, read through 1.1 and 1.2. Parts of Chapter 1.3 will be taken up later. Among the highlights in Chapter 1.3, expressing a linear combination as matrix-vector (or vector-matrix) multiplication is crucial, and is also covered in the lab problems.
To check if a vector is a linear combination of other vectors, we can write out a system of linear equations to solve. The ability to write it this way is crucial. We will emphasize this aspect in the next module, but you probably already see it if you understand how linear combinations are multiplications between matrices and vectors.
Solving linear equations, the section on the Inverse Matrix, Cyclic Differences, Independence and Dependence are not important right away. If you have not seen these concepts before, it is better to skip them for now.
Students often crave visualizations of what is happening, and perhaps it is the only way you think about vectors now. But in a manner of speaking, this is also something to unlearn. It does not mean forgetting about what you know in geometry in 2- or 3-d. Rather it means distilling this into axioms and thinking in terms of axioms (not visualizations).
The reason is that we will be dealing with vectors not with 2 or 3 coordinates, but with large numbers of coordinates (hundreds or thousands in our examples in class, but usually tens of thousands, millions, or billions, or even more in real life). Higher dimensions behave very differently from our intuition or drawings in 2- or 3-d. Just geometry will not get us far.
To make the leap, you have to develop a more axiomatic way of thinking about vectors. It is hard in the beginning, but you must drop the training wheels of literal geometric visualization. Axiomatic rather than geometric thinking is the point of the course, since it turns out to be far more powerful. In this module, we still make a few comments on representations of addition, scaling, and visualization of simple sets of linear combinations. But the emphasis will be on the axioms that will take us quite far.
When you have no recourse to geometry, all you have to fall back on are axioms underlying manipulating vectors and linear combinations. You are very likely to feel uncomfortable with this approach—even more so, if in this module, you do not pay attention to axiomatic vector manipulation.
Once you get a little familiar with axiomatic reasoning, we will replace geometric visualization with a different sort of rendering that captures the essence of this axiomatic approach without being a true geometric representation.