Chapter 11. Introduction to Linear Algebra

Vectors. A vector has magnitude and direction. In the plane, v = (v₁, v₂) has length √(v₁² + v₂²). The dot product is u · v = u₁v₁ + u₂v₂ = |u||v| cos θ, so two vectors are perpendicular when their dot product is zero [14][15].

Matrices. A matrix is a rectangular array of numbers. Matrices of the same size add entry by entry. The product AB is defined when the number of columns of A equals the number of rows of B, and in general AB ≠ BA.

Determinant and inverse of a 2 × 2 matrix. For A = [[a, b], [c, d]], det A = ad − bc. If det A ≠ 0, the inverse is A⁻¹ = (1/det A) [[d, −b], [−c, a]].

Linear systems. A system can be written Ax = b. It has a unique solution if A is invertible. Gaussian elimination solves systems of any size by row operations [14].

Eigenvalues. A non-zero vector v is an eigenvector of A with eigenvalue λ if Av = λv. Eigenvalues are found from det(A − λI) = 0 [14][15].

Review: Find det of [[1, 2], [3, 4]]. (Answer: 1·4 − 2·3 = −2.)