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A scalar is a single real or complex number, the simplest object in linear algebra. Building from it, a vector is an ordered list of scalars, which can be visualized geometrically...
A matrix is a rectangular array of numbers. Its transpose \(A^T\) is obtained by swapping rows and columns, so \((A^T)_{ij} = A_{ji}\). The inverse \(A^{-1}\) of a square matrix is...
The determinant is a scalar that captures key information about a square matrix. For the 2×2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) it is simply \(ad - bc\). Geome...
An eigenvalue of a square matrix \(A\) is a scalar \(\lambda\) for which \(Av = \lambda v\) for some nonzero vector \(v\); the corresponding vector \(v\) is the eigenvector. Eigenv...
The inner (dot) product of two real vectors \(u\) and \(v\) is \(u \cdot v = \sum_i u_i v_i = u^T v\); it equals zero precisely when \(u\) and \(v\) are orthogonal. From it comes t...
Factorizations express complicated matrices as products of simpler ones, exposing structure and enabling efficient computation. The LU decomposition writes a square matrix as \(A =...
For systems of linear equations, row operations — swapping two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another — transform a matrix without...