Vectors

In school, you learned how to plot points on “math paper”. Here are two example points:

We call the horizontal line $x$ and the vertical line $y$. We draw a number line on each of them. Then we can describe any point on the paper using the point's position on each line. The red point has $x=2$ and $y=3$. What is the $x$ value of the blue point?

If a point has $x=2$ and $y=3$, a mathematician might write that the point is $(2,3)$, or $[\begin{array}{c}2\\ 3\end{array}]$. But we will write the point in Python notation: [2,3]. We call this object [2,3] a vector.

Saying p = [2,3] is nicer than saying px = 2 and py = 3. Everything about the point is packaged in one object, and we don't have to make up names like x and y.

So far, we've been describing 2D space. The kind of space that old arcade games use. But now imagine a 3D space, like the one you live in. We usually imagine the lines $x$ and $y$ as before, plus a new line called $z$, which is perpendicular to the $x$ and $y$ lines. Imagine a point $q$ in space which has $x=5$, $y=2$, and $z=8$. How would we write this point in vector notation?

Consider the point [1,0,1,0,1]. It “lives” in some space. How many dimensions does that space have?

Mathematicians call this space ${\mathbb{R}}^5$. (Don't take the “to the power of 5” too seriously -- it's just notation.)

Which of these vectors lives in ${\mathbb{R}}^1$?

Now it's your chance to implement “vector-scalar multiplication”. Here's a Python shell for you to use: