The theory defines a matrix-vector product (Ax) within linear algebra as either a weighted sum of column vectors or a set of row vector dot products, subject to conformability conditions where the inner dimensions must match for definedness and resulting in an M-by-1 output. The concept extends formally through Matrix Equations (Ax = b), which characterize solution existence based on whether every right-hand side vector **b** lies within the column space spanned by the columns of matrix A, a condition equivalent to having a pivot position in every row for all possible **b**. This framework establishes necessary and sufficient conditions where linear systems are consistent across any arbitrary target vector.
The theory defines a matrix-vector product (Ax) within linear algebra as either a weighted sum of column vectors or a set of row vector dot products, subject to conformability conditions where the in…