Module 4 Article: Turning Geometry into Arithmetic

The big idea

A matrix is both a grid of numbers and a geometric transformation. The magic of linear algebra is that these two views are the same thing: solving a system of linear equations (arithmetic) is the same as finding the intersection of lines/planes (geometry). This duality is why matrices are everywhere in graphics, statistics, and machine learning.

Matrix operations: the arithmetic view

• Addition: entry-wise, needs same shape.
• Scalar multiplication: multiply every entry.
• Multiplication: c_ij = Σ_k a_ik·b_kj (row of A · column of B).
  • Shape: m×n times n×p = m×p.
  • Not commutative (AB ≠ BA in general).

Worked arithmetic

[[2,1],[1,2]] · [[1],[-1]] = [[2·1+1·(-1)],[1·1+2·(-1)]] = [[1],[-1]]

So M maps the vector (1,−1) to itself — an early sign that (1,−1) is an eigenvector with eigenvalue 1 (Module 6).

The geometric view: matrices as transforms

The columns of a matrix A are the images of the basis vectors:

A = [[2,1],[1,2]]  means  e1=(1,0) → (2,1),   e2=(0,1) → (1,2)

So A skews the unit square into a parallelogram. The area of the image of the unit square is |det(A)|:

det([[2,1],[1,2]]) = 2·2 − 1·1 = 3

The unit square (area 1) becomes a parallelogram of area 3.

Solving linear systems: three faces of one idea

solve Ax = b is equivalent to:

• Algebraically: Gaussian elimination (row reduction).
• Geometrically: find the intersection of hyperplanes.
• Matrix-wise: x = A⁻¹b (if A is invertible).

Worked example: intersect two lines

2x + y = 5      (line 1)
x + 2y = 4      (line 2)
A = [[2,1],[1,2]], b = [5;4]
det(A) = 3 ≠ 0, so A⁻¹ exists.
A⁻¹ = (1/3)·[[2,−1],[−1,2]]
x = A⁻¹ b = (1/3)·[[2,−1],[−1,2]]·[5;4] = (1/3)·[10−4; −5+8] = (1/3)·[6;3] = [2;1]
⇒ x = 2, y = 1.

Check in line 1: 2·2+1 = 5 ✓; line 2: 2 + 2·1 = 4 ✓. The intersection point (2,1) is where the two lines meet.

Determinants: signed volume

det(A) is the signed area/volume scale factor:

• det(AB) = det(A)·det(B) (scaling multiplies).
• det(Aᵀ) = det(A) (volume unchanged by transpose).
• A invertible ⟺ det(A) ≠ 0 (non-zero area ⇒ not squashed flat).
• A negative determinant means the transform flips orientation (a mirror image).

Row operations and the rank

Three elementary row operations (swap, scale, add-multiple) do not change the solution set of a linear system, and they reveal the rank (number of independent rows = number of independent columns). Rank also tells us existence and uniqueness:

System	Meaning
rank(A) = rank(A	b) = n
rank(A) = rank(A	b) < n
rank(A) ≠ rank(A	b)

Common mistakes

• Assuming (A+B)² = A² + 2AB + B² (fails because AB ≠ BA in general).
• Writing (AB)ᵀ = AᵀBᵀ — the order reverses to BᵀAᵀ.
• Thinking det(A+B) = det(A)+det(B) — determinant is not additive.
• Forgetting that A⁻¹ only exists for square, non-singular matrices.

Exam angle

For "solve Ax = b and interpret geometrically":

1. Row-reduce [A | b] to row-echelon form.
2. Back-substitute for the solution.
3. State the geometric interpretation: number of independent equations = rank; solution = intersection point/line/plane; no solution = parallel planes.

Tip: for a 2×2 inverse, memorize [[a,b],[c,d]]⁻¹ = 1/(ad−bc)·[[d,−b],[−c,a]], and always check with A·A⁻¹ = I.

See also

• Module 5: linear transformations as matrices; kernel = null space.
• Module 6: eigenvalues measure how much the transform stretches along eigenvectors.