10.1.1 Example of Matrix-Column-Vector Multiplication via Linear Combinations

#Type/Example #Topic/Linear_Algebra

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Matrix-Column-Vector Multiplication via Linear Combinations

Find the vector $\vec{b} = A \cdot \vec{x}$ where

A = [\begin{matrix} - 3 & 4 & - 3 \\ - 1 & 7 & 6 \\ 0 & 1 & 2 \\ 2 & - 5 & - 2 \end{matrix}], \vec{x} = [\begin{matrix} 5 \\ 2 \\ - 2 \end{matrix}]

Solution. To start, recall our definition of matrix vector multiplication.

[\vec{b}]_{m \times 1} = [A]_{m \times n} \cdot [\vec{x}]_{n \times 1}

Before we can multiply, we need to check if the inner dimensions agree.

A = {[\begin{matrix} - 3 & 4 & - 3 \\ - 1 & 7 & 6 \\ 0 & 1 & 2 \\ 2 & - 5 & - 2 \end{matrix}]}_{4 \times 3}, \vec{x} = {[\begin{matrix} 5 \\ 2 \\ - 2 \end{matrix}]}_{3 \times 1}

The amount of columns of $A$ is $3$ , and the amount of rows of $\vec{x}$ is $3$ . Therefore, the inner dimensions do indeed agree. The size of the product is defined by the outer dimensions, which would be $4 \times 1$ . Let's split $A$ up into 8.13 Column Partition of a Matrix and $\vec{x}$ into individual scalars so we can do 10.1 Matrix-Column-Vector Multiplication via Linear Combinations.

\begin{aligned} {\vec{b}}_{4 \times 1} & = {[\begin{array}{c} - 3 & 4 & - 3 \\ - 1 & 7 & 6 \\ 0 & 1 & 2 \\ 2 & - 5 & - 2 \end{array}]}_{4 \times 3} \cdot {[\begin{array}{c} 5 \\ 2 \\ - 2 \end{array}]}_{3 \times 1} & Determine dimensions \\ = 5 [\begin{array}{c} - 3 \\ - 1 \\ 0 \\ 2 \end{array}] + 2 [\begin{array}{c} 4 \\ 7 \\ 1 \\ - 5 \end{array}] + - 2 [\begin{array}{c} - 3 \\ 6 \\ 2 \\ - 2 \end{array}] & Linear combination \\ = [\begin{array}{c} - 15 \\ - 5 \\ 0 \\ 10 \end{array}] + [\begin{array}{c} 8 \\ 14 \\ 2 \\ - 10 \end{array}] + [\begin{array}{c} 6 \\ - 12 \\ - 4 \\ 4 \end{array}] & Scalar-vector multiplication \\ = {[\begin{array}{c} - 15 + 8 + 6 \\ - 5 + 14 - 12 \\ 0 + 2 - 4 \\ 10 - 10 + 4 \end{array}]}_{4 \times 1} & Vector addition \\ = [\begin{array}{c} - 1 \\ - 3 \\ - 2 \\ 4 \end{array}] = [\vec{b}]_{4 \times 1} = [A]_{4 \times 3} \cdot [\vec{x}]_{3 \times 1} \end{aligned}