Matrix Multiplication as Column Combinations
Multiplying a matrix by a vector constructs a linear combination of the matrix columns weighted by the vector components.
The Column Perspective
When multiplying a matrix A by a column vector x, the result is not just a set of dot products. Instead, it is a single vector formed by taking a linear combination of the columns of A, where the entries of x serve as scalar weights. This structural view clarifies how linear transformations stretch and combine basis vectors.
A \mathbf{x} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_nMatrix Product Columns
Extending this concept to the product of two matrices, A and B, each column of the resulting matrix C is computed by multiplying A by the corresponding column of B. Consequently, every column in C is a linear combination of the columns of A. This guarantees that the column space of the product AB is always contained within the column space of A.
C = A B \implies \mathbf{c}_j = A \mathbf{b}_jSolving Linear Systems
Viewing matrix multiplication through column combinations provides a clear geometric picture of linear systems. The equation Ax = b asks whether the target vector b can be formed using the columns of A. A solution exists if and only if b lies within the span of A's columns, with x containing the required combination weights.
A \mathbf{x} = \mathbf{b} \iff \mathbf{b} \in \text{Col}(A)Flashcard
Matrix-vector product as a column combination
Check your understanding
If matrix A has columns a_1 and a_2, what is the output of the product A multiplied by the vector [3, -2]^T?