Matrix-Vector Multiplication Rules and Dimensions in Matrix Algebra
The core principle establishes a strict dimensional compatibility rule wherein the inner dimensions of interacting matrices must align for linear combination to occur within Euclidean space. This concept defines matrix-vector multiplication formally through tensor contraction operations, specifically mapping vectors from $\mathbb{R}^n$ into transformed vector spaces via linear operators represented by $m \times n$ coefficient arrays. It functions as a fundamental mechanism in the subfield of Linear Algebra, governing how abstract systems transform state vectors while preserving structural integrity under bilinear mappings.
Multiplying a matrix by a column vector | Matrices | Precalculus | Khan Academy
Matrix-vector multiplication is a fundamental linear algebra operation defined by treating a column vector as a matrix with dimensions $n \times 1$. The core mechanism requires the number of columns …