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Linear Algebra Terms
Vectors, matrices and the core ideas of linear algebra.
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- ScalarTermA single number, as opposed to a vector
- VectorTermAn ordered list of numbers; has magnitude and direction
- Magnitude (norm) of a vectorTermIts length, √(x² + y² + ...)
- Unit vectorTermA vector of length 1
- Zero vectorTermThe vector whose entries are all 0
- Vector additionTermAdd matching entries
- Scalar multiplicationTermMultiply every entry of a vector or matrix by the same number
- Dot productTermSum of products of matching entries; a · b = |a||b| cos θ
- Perpendicular vectorsTermTheir dot product is 0 (orthogonal)
- Cross productTermA vector perpendicular to two 3D vectors; length |a||b| sin θ
- Linear combinationTermA sum of vectors each multiplied by a scalar
- SpanTermThe set of all linear combinations of some vectors
- Linear independenceTermNo vector in the set is a linear combination of the others
- Linear dependenceTermAt least one vector can be built from the others
- BasisTermA linearly independent set of vectors that spans a space
- DimensionTermThe number of vectors in a basis
- Standard basis of R²Term(1, 0) and (0, 1)
- Vector spaceTermA set where vectors can be added and scaled by numbers following set rules
- SubspaceTermA vector space inside a larger one
- MatrixTermA rectangular array of numbers in rows and columns
- Order (size) of a matrixTermRows × columns, such as 2 × 3
- Square matrixTermA matrix with the same number of rows and columns
- Identity matrixTermA square matrix with 1s on the main diagonal and 0s elsewhere; AI = A
- Zero matrixTermA matrix with all entries 0
- Diagonal matrixTermA square matrix with zeros off the main diagonal
- Symmetric matrixTermA matrix equal to its transpose
- TransposeTermSwap rows and columns; written Aᵀ
- Matrix additionTermAdd matching entries; matrices need the same size
- Matrix multiplication ruleTermAn m × n matrix times an n × p matrix gives an m × p matrix
- Is matrix multiplication commutative?TermNo. AB is usually not equal to BA
- Inverse matrixTermA⁻¹ such that A A⁻¹ = I
- Invertible (nonsingular) matrixTermA square matrix with an inverse; its determinant is not 0
- Singular matrixTermA matrix with no inverse; determinant 0
- Determinant of a 2×2 matrix [a b; c d]Termad − bc
- Determinant meaningTermScale factor of area or volume under the matrix's transformation
- Trace of a matrixTermThe sum of the diagonal entries
- RankTermThe number of linearly independent rows (or columns)
- Null spaceTermAll vectors x with Ax = 0
- Column spaceTermThe span of the columns of a matrix
- Row spaceTermThe span of the rows of a matrix
- Linear transformationTermA function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u)
- System of linear equations as a matrixTermAx = b
- Augmented matrixTermThe coefficient matrix with the constants column attached
- Gaussian eliminationTermUsing row operations to turn a matrix into row-echelon form
- Row operationsTermSwap rows, multiply a row by a nonzero number, add a multiple of one row to another
- Row-echelon formTermA staircase form with leading entries moving to the right
- Reduced row-echelon formTermRow-echelon form with leading 1s and zeros above and below them
- PivotTermA leading nonzero entry in a row used in elimination
- EigenvalueTermA number λ such that Av = λv for some nonzero vector v
- EigenvectorTermA nonzero vector that only gets scaled by a matrix
- Characteristic polynomialTermdet(A − λI); its roots are the eigenvalues
- Orthogonal matrixTermA square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹
- ProjectionTermThe shadow of one vector onto another
- Rotation matrix in 2DTerm[cos θ, −sin θ; sin θ, cos θ]
- Cramer's ruleTermSolve a system using ratios of determinants
- Orthonormal basisTermA basis of perpendicular unit vectors
- Gram-Schmidt processTermA method to turn a basis into an orthonormal one
- DiagonalizationTermWriting A = PDP⁻¹ with D diagonal
- Singular value decomposition (SVD)TermFactoring a matrix into rotations and stretches: A = UΣVᵀ
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