Matrices

Understand how matrices organise data into rows and columns, how their dimensions are specified, how vectors fit into this framework as one-row or one-column matrices, and how to identify individual entries using subscript notation.

Learn how to add, subtract and scale matrices entrywise, and use these operations to form linear combinations of matrices with the same dimensions.

Learn how to multiply a matrix by a vector by taking a dot product of each row with the vector, understand the necessary dimension requirements for valid multiplication, and see how this operation underpins key transformations and predictions in machine learning models.

Learn how to determine when the product of two matrices is defined, compute the result using the row-by-column dot product rule, identify the dimensions of the resulting matrix, and recognise that matrix multiplication is not commutative.

Learn how to perform the transpose operation on matrices and vectors, understand its key properties (including behaviour under addition, subtraction and multiplication), and see how the transpose links matrix operations with the inner (dot) product in vector spaces.

Understand the identity matrix as the square matrix with ones on the diagonal and zeros elsewhere, which leaves any matrix or vector unchanged when multiplied, analogous to multiplying by one in arithmetic.

Learn how to compute the determinant of a square matrix using specific formulas and cofactor expansion, and interpret its value as indicating both invertibility and the signed area (or volume) scaled by the matrix.

Understand how the column space of a matrix defines the set of possible outputs $A\vec{x}$, how the rank measures the number of linearly independent columns, and how these concepts determine whether a system $A\vec{x} = \vec{b}$ has solutions, how many solutions, and whether a square matrix is invertible.

Learn how the null space of a matrix captures all solutions to $A\vec{x} = \vec{0}$, how it determines the structure of all solutions to $A\vec{x} = \vec{b}$, how to find a basis for the null space by solving a homogeneous system, and how the rank-nullity theorem links the number of independent columns to the number of free variables.

See how multiplying a vector by a matrix performs a linear transformation, such as scaling, rotating or reflecting that vector, and learn to interpret matrix multiplication as reshaping the entire coordinate grid in a consistent, structured way.

Understand the definition of orthogonal matrices, recognise their appearance in rotation and reflection matrices, and see how these matrices preserve vector lengths, dot products, and geometric structure by satisfying $Q^T Q = I$ and having inverses equal to their transposes.