Matrices and Linear Algebra

What this unit covers

The page moves from the basics of matrices to abstract linear algebra. The Matrices mind maps set out size (m x n), addition, subtraction, multiplication, the identity matrix and the null matrix. Further on you will find Gaussian elimination, systems of linear equations, determinants, inverses, vector spaces, linear transformations, eigenvalues and eigenvectors.

Where to start

If matrices are new to you, begin with the Matrices mind map and the Introduction to Linear Algebra mind map. It links linear systems, matrix properties, determinants and inverses in one view. Then try the Matrix Multiplication Questions quiz to check the arithmetic.

1. Read a Matrices mind map for the vocabulary.
2. Work through the Matrices Notes, which explain finding an inverse using the determinant ad-bc.
3. Practise with the Gaussian Elimination quiz and the Row Reduction slides.

Transformations and geometry

The Matrix Transformation (2D) flashcards ask where a matrix sends (1,0) and (0,1), which is a quick way to read a 2x2 matrix. The Invariation Questions slides ask for lines of invariant points. The slides on 3x3 determinants and factorization treat the determinant as a volume scale factor. Several resources mention A-level Further Maths, so school students will find material that suits them.

Vector spaces and beyond

For university work, the Vector Space mind map covers linear independence, span, subspaces, basis and inner product spaces. The Linear Transformation mind map covers the kernel and range. Diagonalization explains when a matrix is similar to a diagonal matrix, and the notes created from mind maps give a written version of the same ideas.

Using quizzes and maps together

Read a mind map first, then take the Linear Algebra Quiz and note which questions you miss. Return to the matching map or flashcard deck for those topics, then test yourself again. Everything here is free to open, so choose a topic and start.