2.2 Matrix Multiplication & Linear Combinations

Kristen Pueschel

2.2 Matrix Multiplication & Linear Combinations

Reading

Work on the Preview assignment and read Chapter 2 Section 2 from Understanding Linear Algebra by David Austin.

Addition of Matrices and Scalar Multiplication:

The matrix vector product.

Definition 2.2.2. The product of a matrix A by a vector y will be the linear combination of the columns of A using the components of y as weights.
If A is an mxn matrix,

A = [{\vec{v}}_{1} {\vec{v}}_{2} \dots {\vec{v}}_{n}]

, then

\vec{x}

must be an n-dimensional vector, and the product

A \vec{x}

will be an m-dimensional vector.

A \vec{x} = A [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{n} \end{matrix}] = [{\vec{v}}_{1} {\vec{v}}_{2} \dots {\vec{v}}_{n}] [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{n} \end{matrix}] = c_{1} {\vec{v}}_{1} + \dots + c_{n} {\vec{v}}_{n}

The linear-combination interpretation of matrix-vector products is important, but we can sometimes save time and space in the computations by computing terms individually.