Linear Algebra with Differential Equations

From vectors and matrices to eigenvalues and systems of ODEs. Click a topic to explore.

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STEP 1VECTORS & VECTOR SPACES

The geometric and algebraic language of linear algebra

Vectors are the fundamental objects of linear algebra. A vector in Rⁿ is an ordered list of n real numbers. Understanding vectors — how to combine them, when they are independent, and how they span a space — is the foundation for everything that follows. The course begins here (Sections 2.1–2.7) after the mechanics of linear systems.

A vector v in Rⁿ is a column of n real numbers. The standard operations are addition and scalar multiplication. The dot product gives us a way to measure angles and orthogonality.

Vector addition and scalar multiplicationComponent-wise operations

For u, v in Rⁿ and scalar c: addition is component-wise and scalar multiplication scales every entry. These two operations satisfy the 8 vector space axioms.

\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ \vdots \\ u_n + v_n \end{pmatrix}, \quad c\mathbf{v} = \begin{pmatrix} cv_1 \\ \vdots \\ cv_n \end{pmatrix}

Dot productMeasures alignment and orthogonality

The dot product of two vectors is the sum of products of corresponding components. When the dot product is zero, the vectors are orthogonal (perpendicular).

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

Norm (length) of a vectorEuclidean distance from origin

The norm is the square root of the dot product of a vector with itself. A unit vector has norm 1. To normalise: divide by the norm.

\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Cross product (R³ only)Produces a perpendicular vector

The cross product of two vectors in R³ produces a vector perpendicular to both. Its magnitude equals the area of the parallelogram spanned by the two vectors.

\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

MA1513 — Quick Check

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Q1.A square matrix A is invertible if and only if: