An Intuitive Guide to Linear Algebra

linear-algebramatriceseigenvectorsdeterminantsintuition

Abstraction: Spreadsheet-metaphor intuition for matrices, transformations, and eigenvectors

Key points:

Matrices are "mini-spreadsheets": inputs in vertical columns, operations (linear combinations) in horizontal rows; outputs produced by "pouring" inputs through operations
Linear operations satisfy F(2x) = 2F(x) and F(a+b) = F(a)+F(b); plain scaling by constant is linear, but x+3 and x^2 are not
Matrix multiplication composes transformations: T*N means "apply N then T"; the result is a new matrix encoding both steps
Determinant = size of the output transformation volume; determinant of 0 means the transformation destroys information and is non-invertible
Eigenvector: direction unchanged by the matrix transformation; eigenvalue: the scalar scaling factor along that direction (e.g., globe rotation — poles don't move)
Adding a dummy input of 1 (homogeneous coordinates) enables affine addition constants within the purely multiplicative linear framework

Connections: Betterexplained · Linear Algebra · Matrix Operations · Eigenvectors · Linear Transformations