mathFOLLOW AN IDEA
Linear algebra
Connect vectors, linear transformations and matrix computations, with geometric and algebraic viewpoints.
Vectors and spaces
Vectors
Vectors can be added and scaled; coordinate lists represent vectors relative to a choice of basis.
No reviewed resource yetSpan
The span of a set consists of all finite linear combinations of its vectors.
No reviewed resource yetLinear independence
A collection is linearly independent when the zero vector has only the trivial linear combination using that collection.
No reviewed resource yetBasis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.
No reviewed resource yetVector spaces
A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.
No reviewed resource yet
Matrices and equations
Matrices
A matrix is a rectangular array representing data or a linear map in chosen bases; matrix multiplication represents composition when dimensions agree.
1 reviewed resourcesLinear systems
A system Ax=b asks whether b is a linear combination of the columns of A and, if so, with what coefficients.
No reviewed resource yetGaussian elimination
Elementary row operations simplify a system while preserving its solution set, revealing pivots and free variables.
No reviewed resource yetRank and nullity
The rank is the dimension of the image; rank plus nullity equals the dimension of the domain for a finite-dimensional linear map.
No reviewed resource yetDeterminants
The determinant of a square matrix tracks signed volume scaling and is nonzero exactly when the matrix is invertible.
1 reviewed resources
Transformations and spectra
Linear transformations
A linear transformation preserves vector addition and scalar multiplication; its matrix depends on the chosen bases.
No reviewed resource yetEigenvalues and eigenvectors
A nonzero eigenvector is scaled by a linear operator; the scale factor is its eigenvalue.
No reviewed resource yetDiagonalization
A matrix is diagonalizable when it has a basis of eigenvectors; having repeated eigenvalues does not by itself decide diagonalizability.
No reviewed resource yet
Orthogonality
Inner products
An inner product defines lengths and orthogonality through a positive-definite bilinear or sesquilinear structure.
No reviewed resource yetOrthogonal projections
Orthogonal projection splits a vector into a component in a subspace and a component perpendicular to that subspace.
No reviewed resource yetLeast squares
Least squares minimizes the squared residual norm; a minimizer corresponds to projecting the target onto the column space.
No reviewed resource yet