Linear Maps — Topic Summaries
AI-powered summaries of 18 videos about Linear Maps.
18 summaries
Linear Algebra 35 | Rank-Nullity Theorem
Rank–nullity theorem is the organizing rule behind how linear maps “trade” dimensions: for any linear map represented by a matrix with n columns, the...
Linear Algebra 34 | Range and Kernel of a Matrix
Range and kernel are the two core subspaces that determine whether a linear system has solutions and whether those solutions are unique. For an m×n...
Linear Algebra 14 | Column Picture of the Matrix-Vector Product
A matrix can be understood as a “machine” that outputs a vector built from its own columns: multiplying a matrix A by a vector x produces a result Ax...
Linear Algebra 20 | Linear maps induce matrices
Every linear map between finite-dimensional real vector spaces can be turned into a unique matrix—so the abstract action of a function becomes a...
Linear Algebra 29 | Identity and Inverses
Identity matrices and inverses sit at the center of linear algebra because they formalize “do nothing” and “undo what a transformation does.” An n×n...
Abstract Linear Algebra 7 | Change of Basis
Change of basis is the mechanism for translating the same vector in a finite-dimensional vector space between two different coordinate systems. Since...
Linear Algebra 19 | Matrices induce linear maps [dark version]
A matrix doesn’t just store numbers—it automatically defines a linear map between vector spaces, and the usual matrix-vector multiplication is...
Linear Algebra 14 | Column Picture of the Matrix-Vector Product [dark version]
A matrix can be understood as a “column machine” that turns an input vector into an output vector by forming a linear combination of its columns....
Linear Algebra 34 | Range and Kernel of a Matrix [dark version]
Range and kernel turn a matrix into two practical “tests” for solving linear systems: whether a right-hand side can be reached at all, and whether...
Linear Algebra 20 | Linear maps induce matrices [dark version]
Every linear map between finite-dimensional vector spaces can be turned into a unique matrix, and that matrix is determined entirely by what the map...
Abstract Linear Algebra 28 | Equivalent Matrices
Equivalent matrices capture when two different matrix representations actually describe the same linear transformation, even after changing the bases...
Linear Algebra 35 | Rank-Nullity Theorem [dark version]
Rank–nullity theorem is the organizing rule behind how linear maps “trade” dimensions: for any linear map (equivalently, any matrix) from an...
Abstract Linear Algebra 7 | Change of Basis [dark version]
Change of basis is the mechanism for translating the same abstract vector’s coordinates when the underlying basis in a finite-dimensional vector...
Abstract Linear Algebra 22 | Linear Maps
A linear map is defined by two rules—preserving vector addition and scalar multiplication—and that constraint sharply limits what it can do to...
Linear Algebra 54 | Characteristic Polynomial [dark version]
Eigenvalues can be found by turning a matrix problem into a single polynomial equation: for a square matrix A, the eigenvalues are exactly the zeros...
Linear Algebra 29 | Identity and Inverses [dark version]
Identity matrices and matrix inverses are the backbone of turning linear maps into something you can compute—and back again. An n×n identity matrix,...
Abstract Linear Algebra 23 | Combinations of Linear Maps
Linear maps aren’t just single functions between vector spaces—they form their own vector space under addition and scalar multiplication. Given two...
Abstract Linear Algebra 34 | Eigenvalues and Eigenvectors for Linear Maps
Eigenvectors and eigenvalues for a linear map are defined by a simple “scaling” condition: a nonzero vector X is an eigenvector of L if L(X) lands in...