Seeing the algebra: why pictures first

There is a specific failure mode in self-taught technical learning: you collect formulas, learn to manipulate them, and still cannot tell whether an answer is plausible. The fix is not "study more maths" — it is attaching a picture to every symbol before you move on.

Concretely, here is what "seeing" means for the four operations that show up everywhere:

Dot product `a·b` — measures alignment. If `b` is a unit vector, `a·b` is the signed scalar projection of `a` onto `b`; for a non-unit `b`, divide by `|b|`.
Matrix × vector — `A` moves `x` somewhere; the column of `A` where `x = (0,0,…,1,…)` lands tells you what that basis direction became.
Determinant — `|det(A)|` is the area (or volume) scale factor; a negative determinant also flips orientation, and zero collapses dimension.
Eigenvector — a real eigenvector stays on its line under `A`; its eigenvalue is the scale factor and may be negative, which reverses direction.

# The single highest-value exercise in this whole layer: watch it happen.
import numpy as np, matplotlib.pyplot as plt

A = np.array([[2.0, 1.0],
              [0.0, 1.5]])       # we will keep this time

# 1) where do the basis vectors land?
print("e1 ->", A @ [1, 0])       # the FIRST COLUMN of A
print("e2 ->", A @ [0, 1])       # the SECOND COLUMN of A

# 2) area scaling = determinant
print("det  ", np.linalg.det(A)) # 3.0  -> areas triple

# 3) directions A does not rotate
w, V = np.linalg.eig(A)
print("eigenvalues ", w)         # 2.0 and 1.5
print("eigenvectors", V)         # columns: directions that only stretch

# 4) draw a unit square; draw where A sends it -> you can SEE det as area.