#separator:tab #html:false #tags column:4 Scalar A single number, as opposed to a vector math-linear-algebra Vector An ordered list of numbers; has magnitude and direction math-linear-algebra Magnitude (norm) of a vector Its length, √(x² + y² + ...) math-linear-algebra Unit vector A vector of length 1 math-linear-algebra Zero vector The vector whose entries are all 0 math-linear-algebra Vector addition Add matching entries math-linear-algebra Scalar multiplication Multiply every entry of a vector or matrix by the same number math-linear-algebra Dot product Sum of products of matching entries; a · b = |a||b| cos θ math-linear-algebra Perpendicular vectors Their dot product is 0 (orthogonal) math-linear-algebra Cross product A vector perpendicular to two 3D vectors; length |a||b| sin θ math-linear-algebra Linear combination A sum of vectors each multiplied by a scalar math-linear-algebra Span The set of all linear combinations of some vectors math-linear-algebra Linear independence No vector in the set is a linear combination of the others math-linear-algebra Linear dependence At least one vector can be built from the others math-linear-algebra Basis A linearly independent set of vectors that spans a space math-linear-algebra Dimension The number of vectors in a basis math-linear-algebra Standard basis of R² (1, 0) and (0, 1) math-linear-algebra Vector space A set where vectors can be added and scaled by numbers following set rules math-linear-algebra Subspace A vector space inside a larger one math-linear-algebra Matrix A rectangular array of numbers in rows and columns math-linear-algebra Order (size) of a matrix Rows × columns, such as 2 × 3 math-linear-algebra Square matrix A matrix with the same number of rows and columns math-linear-algebra Identity matrix A square matrix with 1s on the main diagonal and 0s elsewhere; AI = A math-linear-algebra Zero matrix A matrix with all entries 0 math-linear-algebra Diagonal matrix A square matrix with zeros off the main diagonal math-linear-algebra Symmetric matrix A matrix equal to its transpose math-linear-algebra Transpose Swap rows and columns; written Aᵀ math-linear-algebra Matrix addition Add matching entries; matrices need the same size math-linear-algebra Matrix multiplication rule An m × n matrix times an n × p matrix gives an m × p matrix math-linear-algebra Is matrix multiplication commutative? No. AB is usually not equal to BA math-linear-algebra Inverse matrix A⁻¹ such that A A⁻¹ = I math-linear-algebra Invertible (nonsingular) matrix A square matrix with an inverse; its determinant is not 0 math-linear-algebra Singular matrix A matrix with no inverse; determinant 0 math-linear-algebra Determinant of a 2×2 matrix [a b; c d] ad − bc math-linear-algebra Determinant meaning Scale factor of area or volume under the matrix's transformation math-linear-algebra Trace of a matrix The sum of the diagonal entries math-linear-algebra Rank The number of linearly independent rows (or columns) math-linear-algebra Null space All vectors x with Ax = 0 math-linear-algebra Column space The span of the columns of a matrix math-linear-algebra Row space The span of the rows of a matrix math-linear-algebra Linear transformation A function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u) math-linear-algebra System of linear equations as a matrix Ax = b math-linear-algebra Augmented matrix The coefficient matrix with the constants column attached math-linear-algebra Gaussian elimination Using row operations to turn a matrix into row-echelon form math-linear-algebra Row operations Swap rows, multiply a row by a nonzero number, add a multiple of one row to another math-linear-algebra Row-echelon form A staircase form with leading entries moving to the right math-linear-algebra Reduced row-echelon form Row-echelon form with leading 1s and zeros above and below them math-linear-algebra Pivot A leading nonzero entry in a row used in elimination math-linear-algebra Eigenvalue A number λ such that Av = λv for some nonzero vector v math-linear-algebra Eigenvector A nonzero vector that only gets scaled by a matrix math-linear-algebra Characteristic polynomial det(A − λI); its roots are the eigenvalues math-linear-algebra Orthogonal matrix A square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹ math-linear-algebra Projection The shadow of one vector onto another math-linear-algebra Rotation matrix in 2D [cos θ, −sin θ; sin θ, cos θ] math-linear-algebra Cramer's rule Solve a system using ratios of determinants math-linear-algebra Orthonormal basis A basis of perpendicular unit vectors math-linear-algebra Gram-Schmidt process A method to turn a basis into an orthonormal one math-linear-algebra Diagonalization Writing A = PDP⁻¹ with D diagonal math-linear-algebra Singular value decomposition (SVD) Factoring a matrix into rotations and stretches: A = UΣVᵀ math-linear-algebra