Chapter 2 - Linear Combinations, Span, and Basis Vectors

Assuming you are already familiar with vectors or looked at the previous chapter…

To frame the idea of a vector for this content, it is best for you to think of coordinates as a scalar, meaning think how each values stretches or squishes vectors…

For this purpose, let’s familiarize ourselves with two very important vectors.

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Unit Vectors

The first one to know is the unit vector pointing to the right with a length of 1:

$$\hat{i}$$

And the second one is the unit vector pointing straight up, also with a length of 1:

$$\hat{j}$$

These are the unit vectors for the $x$ and $y$ coordinates. Another way to think of them is as their matrix notation counterparts:

$$\hat{i}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\hat{j}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$$