10.4.1 Example of Row Vector Matrix Multiplication via Linear Combinations

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Row Vector Matrix Multiplication via Linear Combinations

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

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

Solution. To start, let's recall our definition of 10.4 Row-Vector-Matrix Multiplication via Linear Combinations.

[\vec{b}]_{1 \times n} = [{\vec{x}}^{T}]_{1 \times 4} \cdot [A]_{4 \times 3}

Because our modeling matrix shows up on the right, we're going to slice our matrix into row partitions. We then take the 6.1 Linear Combination of Vectors.

\begin{aligned} \vec{b} & = {\vec{x}}^{T} A \\ = x_{1} A (1, :) + x_{2} A (2, :) + x_{3} A (3, :) + x_{4} A (4, :) \\ = x_{1} [\begin{array}{c} a_{11} & a_{12} & a_{13} \end{array}] \\ + x_{2} [\begin{array}{c} a_{21} & a_{22} & a_{23} \end{array}] \\ + x_{3} [\begin{array}{c} a_{31} & a_{32} & a_{33} \end{array}] \\ + x_{4} [\begin{array}{c} a_{41} & a_{42} & a_{43} \end{array}] \\ = (2) [\begin{array}{c} - 3 & 4 & - 3 \end{array}] \\ + (- 1) [\begin{array}{c} - 1 & 7 & 6 \end{array}] \\ + (5) [\begin{array}{c} 0 & 1 & 2 \end{array}] \\ + (3) [\begin{array}{c} 2 & - 5 & - 2 \end{array}] \\ = [\begin{array}{c} - 6 & 8 & - 6 \end{array}] \\ + [\begin{array}{c} 1 & - 7 & - 6 \end{array}] \\ + [\begin{array}{c} 0 & 5 & 10 \end{array}] \\ + [\begin{array}{c} 6 & - 15 & - 6 \end{array}] \\ = [\begin{array}{c} 1 & - 9 & - 8 \end{array}] \end{aligned}