About this topic
Before you begin
Solve linear systems by row reduction, represent linear transformations, work with linear spaces, rank and nullity, and diagonalise matrices.
Learning goals
- Solve linear systems by row reduction and interpret the solution set geometrically
- Calculate determinants and inverses of 2 × 2 and 3 × 3 matrices
- Find rank, null space and column space and relate rank to nullity
- Find real eigenvalues and their eigenvectors of 2 × 2 and 3 × 3 matrices
- Diagonalise a matrix as QDQ⁻¹ and use the form to find powers
- Represent and interpret linear transformations by matrices
- Test subspaces, linear independence and span, and find a basis and dimension
Syllabus statements covered
- Include: use of matrices to represent a set of linear equations; operations on 3 × 3 matrices; determinant of a square matrix and inverse of a non-singular matrix (2 × 2 and 3 × 3 matrices only); use of matrices to solve a set of linear equations (including row reduction and echelon forms, and geometrical interpretation of the solution); linear transformations and matrices from ℝⁿ → ℝᵐ, where n, m ⩽ 3; eigenvalues and eigenvectors of square matrices (2 × 2 and 3 × 3 matrices, restricted to cases where the eigenvalues are real); diagonalisation of a square matrix M by expressing the matrix in the form QDQ⁻¹, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use of this expression such as to find the powers of M; linear spaces and subspaces, and the axioms (restricted to spaces of finite dimension over the field of real numbers only); linear independence and span; basis and dimension (in simple cases), including use of terms such as ‘column space’, ‘row space’, ‘range space’ and ‘null space’; rank of a square matrix and relation between rank, dimension of null space and order of the matrix.
Choose a lesson
Work through them in order.
- Not startedRow reduction, rank and null spaceNextSolve linear systems by row reduction, interpret unique, infinite and empty solution sets, find determinants and inverses, and relate rank to the null and column spaces.
- Not startedEigenvalues, eigenvectors and diagonalisationNextFind the directions a matrix only stretches, diagonalise it as QDQ⁻¹ and use that form to find its powers.
- Not startedLinear transformations and linear spacesNextRepresent linear transformations by matrices, find null spaces and ranges, and test subspaces, independence, span and bases.