18 lessons across 4 weeks, 75 verified past-year questions. Free preview below — sign in for the full interactive lessons, AI tutor, and mock exams.
Week 1
- Vectors — A vector is a list of numbers (row or column) that supports component-wise addition and scalar multiplication, and can be visualized as an arrow from the origin in coordinate space.
- Matrices — A matrix is a rectangular array of numbers arranged in rows and columns that supports operations like addition, scalar multiplication, and matrix multiplication, and can represent systems of linear equations.
- Systems of Linear Equations — A system of linear equations can be represented as a matrix equation Ax = b, and it can have no solution, exactly one unique solution, or infinitely many solutions.
- Determinants — The determinant is a scalar value associated with a square matrix, computed inductively using minors and cofactors, that satisfies key properties regarding matrix products, row operations, and transposes.
Week 2
- Properties of Determinants — Learn how to compute determinants by expanding along any row or column, and understand key properties including the effect of row and column operations on the determinant.
- Cramer's Rule — Use determinants to solve systems of linear equations with an invertible coefficient matrix by replacing columns with the constant vector and taking ratios of determinants.
- Matrix Inverse via Adjugate — Understand the uniqueness and existence of the matrix inverse, and compute it using the adjugate matrix divided by the determinant when the determinant is non-zero.
- Row Echelon Form — Identify row echelon form and reduced row echelon form, and use them to easily read off solutions to systems of linear equations by distinguishing dependent and independent variables.
- Row Reduction — Apply elementary row operations to systematically reduce any matrix to its reduced row echelon form, and use this process to efficiently compute determinants.
- Gaussian Elimination Method — Solve arbitrary systems of linear equations by forming the augmented matrix, applying row reduction, and reading off all solutions or detecting inconsistency, including analysis of homogeneous systems.
Week 3
- Vector Spaces — A vector space is a set equipped with addition and scalar multiplication operations that satisfy eight specific axioms, generalizing the familiar properties of vectors in $\mathbb{R}^n$.
- Linear Dependence — A set of vectors is linearly dependent if the zero vector can be expressed as a linear combination of them using coefficients that are not all zero.
- Linear Independence — A set of vectors is linearly independent if the only way their linear combination equals the zero vector is when all scalar coefficients are zero.
Week 4
- Span and Spanning Sets — The span of a set of vectors is the subspace of all possible linear combinations of those vectors, and a spanning set is a set whose linear combinations can produce any vector in the entire vector space.
- Definition of a Basis — A basis for a vector space is a linearly independent set that also spans the space, striking an optimal balance between being large enough to span and small enough to avoid linear dependence.
- Equivalent Conditions and Methods for Finding a Basis — A basis can be characterized as a maximal linearly independent set or a minimal spanning set, and can be found algorithmically either by appending linearly independent vectors or by deleting linearly dependent vectors from a spanning set.
- Dimension and Rank of a Vector Space — The dimension (or rank) of a vector space is the number of vectors in any basis for that space, a well-defined quantity because all bases of a vector space have the same size.
- Computing Basis and Dimension Using Gaussian Elimination — You can systematically find the dimension and a basis for a vector space by applying Gaussian elimination to a matrix, either using the row method to get new basis vectors or the column method to select basis vectors from the original spanning set.
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