Scalar: A single number, as opposed to a vector
Vector: An ordered list of numbers; has magnitude and direction
Magnitude (norm) of a vector: Its length, √(x² + y² + ...)
Unit vector: A vector of length 1
Zero vector: The vector whose entries are all 0
Vector addition: Add matching entries
Scalar multiplication: Multiply every entry of a vector or matrix by the same number
Dot product: Sum of products of matching entries; a · b = |a||b| cos θ
Perpendicular vectors: Their dot product is 0 (orthogonal)
Cross product: A vector perpendicular to two 3D vectors; length |a||b| sin θ
Linear combination: A sum of vectors each multiplied by a scalar
Span: The set of all linear combinations of some vectors
Linear independence: No vector in the set is a linear combination of the others
Linear dependence: At least one vector can be built from the others
Basis: A linearly independent set of vectors that spans a space
Dimension: The number of vectors in a basis
Standard basis of R²: (1, 0) and (0, 1)
Vector space: A set where vectors can be added and scaled by numbers following set rules
Subspace: A vector space inside a larger one
Matrix: A rectangular array of numbers in rows and columns
Order (size) of a matrix: Rows × columns, such as 2 × 3
Square matrix: A matrix with the same number of rows and columns
Identity matrix: A square matrix with 1s on the main diagonal and 0s elsewhere; AI = A
Zero matrix: A matrix with all entries 0
Diagonal matrix: A square matrix with zeros off the main diagonal
Symmetric matrix: A matrix equal to its transpose
Transpose: Swap rows and columns; written Aᵀ
Matrix addition: Add matching entries; matrices need the same size
Matrix multiplication rule: An m × n matrix times an n × p matrix gives an m × p matrix
Is matrix multiplication commutative?: No. AB is usually not equal to BA
Inverse matrix: A⁻¹ such that A A⁻¹ = I
Invertible (nonsingular) matrix: A square matrix with an inverse; its determinant is not 0
Singular matrix: A matrix with no inverse; determinant 0
Determinant of a 2×2 matrix [a b; c d]: ad − bc
Determinant meaning: Scale factor of area or volume under the matrix's transformation
Trace of a matrix: The sum of the diagonal entries
Rank: The number of linearly independent rows (or columns)
Null space: All vectors x with Ax = 0
Column space: The span of the columns of a matrix
Row space: The span of the rows of a matrix
Linear transformation: A function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u)
System of linear equations as a matrix: Ax = b
Augmented matrix: The coefficient matrix with the constants column attached
Gaussian elimination: Using row operations to turn a matrix into row-echelon form
Row operations: Swap rows, multiply a row by a nonzero number, add a multiple of one row to another
Row-echelon form: A staircase form with leading entries moving to the right
Reduced row-echelon form: Row-echelon form with leading 1s and zeros above and below them
Pivot: A leading nonzero entry in a row used in elimination
Eigenvalue: A number λ such that Av = λv for some nonzero vector v
Eigenvector: A nonzero vector that only gets scaled by a matrix
Characteristic polynomial: det(A − λI); its roots are the eigenvalues
Orthogonal matrix: A square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹
Projection: The shadow of one vector onto another
Rotation matrix in 2D: [cos θ, −sin θ; sin θ, cos θ]
Cramer's rule: Solve a system using ratios of determinants
Orthonormal basis: A basis of perpendicular unit vectors
Gram-Schmidt process: A method to turn a basis into an orthonormal one
Diagonalization: Writing A = PDP⁻¹ with D diagonal
Singular value decomposition (SVD): Factoring a matrix into rotations and stretches: A = UΣVᵀ

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