Formula sheet

Vectors and matrices — Formula Sheet

7 formulas from Linear Algebra, each with what it means, when to use it and the trap it sets.

u \cdot v = \sum u_{i} v_{i} = ∣ u ∣∣ v ∣ cos θ

What it says: How much one vector points along another, as a single number.
When to reach for it: Angles between vectors, testing perpendicularity, and projections.
Watch out: The result is a scalar, not a vector. Zero means perpendicular, not "no answer".

(1, 2) \cdot (3, - 1) = 3 - 2 = 1

Positive, so the two vectors point broadly the same way.

∣ v ∣ = v \cdot v = v_{1}^{2} + \dots + v_{n}^{2}

What it says: Pythagoras, in as many dimensions as you like.
When to reach for it: Normalising a vector, or any distance in space.
Watch out: A unit vector is v divided by its length — dividing by the length squared is the standard slip.

∣ (3, 4) ∣ = 5

So the unit vector in that direction is (3/5, 4/5).

(A B)_{ij} = k \sum A_{ik} B_{k j}

What it says: Entry i,j is row i of A dotted with column j of B.
When to reach for it: Composing two transformations, or applying a matrix to a vector.
Watch out: Inner dimensions must match: (m×n)(n×p) gives m×p. And AB is generally not BA — matrix multiplication does not commute.

A_{2 \times 3} B_{3 \times 1} = C_{2 \times 1}

Check the shapes before computing anything.

det (a c b d) = a d - b c

What it says: The signed factor by which the matrix scales area.
When to reach for it: Testing invertibility, and as the denominator of the 2×2 inverse.
Watch out: Determinant zero means singular — no inverse, and the transformation flattens space.

det (2412) = 0

The rows are proportional, so the matrix is not invertible.

A^{- 1} = \frac{1}{a d - b c} (d - c - b a)

What it says: Swap the diagonal, negate the off-diagonal, divide by the determinant.
When to reach for it: Solving Ax = b for a small system, or undoing a transformation.
Watch out: Inverting a matrix is nothing like inverting its entries. And it fails outright when det A = 0.

(1324)^{- 1} = \frac{1}{- 2} (4 - 3 - 2 1)

Always verify by checking that A⁻¹A gives the identity.

proj_{u} v = \frac{v \cdot u}{u \cdot u} u

What it says: The shadow of v cast onto the line through u.
When to reach for it: Least squares, Gram–Schmidt, and splitting a vector into components.
Watch out: The denominator u·u disappears only when u is already a unit vector. Dropping it otherwise scales the answer wrongly.

proj_{(1, 0)} (3, 4) = (3, 0)

What is left, (0,4), is perpendicular to u — always check that.

A v = λ v ⟺ det (A - λ I) = 0

What it says: Directions the matrix only stretches, and the factor by which it stretches them.
When to reach for it: Stability, principal axes, diagonalisation, and systems of differential equations.
Watch out: Subtract λ from the diagonal only. Then solve (A−λI)v = 0 for each λ to get its eigenvectors.

(2003) : λ = 2, 3

A diagonal matrix wears its eigenvalues on its diagonal.

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