Linear Algebra: Introduction to Matrices

These flashcards cover fundamental concepts about matrices, including definitions, operations, and special types of matrices in linear algebra.

Matrix

A rectangular array of numbers, symbols, or expressions arranged in rows and columns.

Dimensions of a Matrix

Described as m × n, where m represents the number of rows and n the number of columns.

Subscript Notation

Used to refer to individual elements within a matrix, denoted as aij for the element in the i-th row and j-th column.

Transpose of a Matrix

An operation that interchanges the rows and columns of a matrix, denoted as AT.

Matrix Addition & Subtraction

Operations that require matrices to have the same dimensions and are performed element-wise.

Scalar Multiplication

The operation of multiplying each element of a matrix by a scalar (a single number).

Matrix Multiplication

A complex operation where the product of two matrices is determined by the dot product of their rows and columns.

Dot Product

The sum of the products of the corresponding entries of two sequences of numbers, used in matrix multiplication.

Elementary Row Operations

Procedures applied to augment matrices, including swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.

Determinant

A scalar value that provides information about a square matrix, denoted as det(A) or |A|.

Invertible Matrix

A square matrix that has an inverse, such that AA−1 = I, where I is the identity matrix.

Symmetric Matrix

A square matrix that is equal to its transpose, meaning aij = aji for all elements.

Orthogonal Matrix

A square matrix whose transpose is equal to its inverse, implying QT = Q−1.

Idempotent Matrix

A square matrix A such that A2 = A.

Nilpotent Matrix

A square matrix A for which some positive integer power of the matrix is the zero matrix.

Gaussian Elimination

A systematic method used to solve systems of linear equations by transforming a matrix into its echelon form.

Matrix-Vector Product

Multiplying a matrix by a vector, resulting in a new vector; used to represent systems of linear equations.

Row Vector

A matrix with a single row, denoted as 1 × n, where n is the number of columns.

Column Vector

A matrix with a single column, denoted as n × 1, where n is the number of rows.

Outer Product

The product of a column vector by a row vector, resulting in a matrix.

Eigenvalues

Scalars associated with a linear transformation represented by a matrix, significant in various advanced mathematical concepts.

Linear Transformation

A mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Projection Matrix

A matrix that maps a vector onto a subspace, such as onto a line or plane.