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, [latex]A= [\vec{v}_1~ \vec{v}_2 ~\dots~ \vec{v}_n][/latex], then [latex]\vec{x}[/latex] must be an n-dimensional vector, and the product [latex]A\vec{x}[/latex] will be an m-dimensional vector.
[latex]A\vec{x} =A\begin{bmatrix} c_1 \\ c_2\\ \vdots \\ c_n \end{bmatrix}= [\vec{v}_1~ \vec{v}_2 ~\dots~ \vec{v}_n]\begin{bmatrix} c_1 \\ c_2\\ \vdots \\ c_n \end{bmatrix} = c_1 \vec{v}_1+\dots + c_n \vec{v}_n[/latex]
If A is an mxn matrix, [latex]A= [\vec{v}_1~ \vec{v}_2 ~\dots~ \vec{v}_n][/latex], then [latex]\vec{x}[/latex] must be an n-dimensional vector, and the product [latex]A\vec{x}[/latex] will be an m-dimensional vector.
[latex]A\vec{x} =A\begin{bmatrix} c_1 \\ c_2\\ \vdots \\ c_n \end{bmatrix}= [\vec{v}_1~ \vec{v}_2 ~\dots~ \vec{v}_n]\begin{bmatrix} c_1 \\ c_2\\ \vdots \\ c_n \end{bmatrix} = c_1 \vec{v}_1+\dots + c_n \vec{v}_n[/latex]
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.
Matrix equations.
Example 1 uses the definition of the matrix vector product
In Example 2 we see how to translate from a matrix equation to an augmented matrix.
Matrix Multiplication:
