Why vectors show up so early

EECS gets abstract fast.

The reason vectors appear everywhere is that they let us encode many quantities at once: position, velocity, features of a data point, coefficients of a polynomial, even probabilities in a finite system.

The mathematical object

can be read as a point, an arrow, or just an ordered pair of numbers, depending on context.

Linear combinations

If

then every vector of the form

is a linear combination of $u$ and $w$.

In the standard basis,

That sounds formal, but it is really just a recipe for building one object from simpler building blocks.

A Python representation

For a first pass, a vector can just be a tuple.

u = (1, 0)
w = (0, 1)

def add(a, b):
    return (a[0] + b[0], a[1] + b[1])

def scale(c, v):
    return (c * v[0], c * v[1])

v = add(scale(3, u), scale(-2, w))
print(v)  # (3, -2)

The code is tiny, but the conceptual point is big:

• a vector is stored as data,
• addition and scalar multiplication become operations,
• mathematics becomes executable.

A tiny graph

This is the sort of thing I mean by “notes on the website” instead of only shipping files away as PDFs.

For the quadratic

you can already read off two roots:

That kind of quick visual feedback is useful even in a theory-heavy course.

Recitation prompts

1. Write a function that computes the dot product of two vectors in $\mathbb{R}^2$.
2. Verify in code that $u \cdot v = v \cdot u$ for several examples.
3. Pick two vectors and decide whether one is a scalar multiple of the other.

Homework direction

Try to answer this in both math language and code language:

If

what does it mean for $v$ to lie on the line $y = 2x$?

In coordinates, the condition is

In code, it becomes a Boolean test.

def on_line(v):
    x, y = v
    return y == 2 * x

That translation from geometry to algebra to code is basically the entire spirit of the course.

If the notes keep that translation visible, the course page is doing its job.