Matrix Multiplication as a Row/Column Operation
Table of contents
1. Left Multiplication as a Row Operation
\[\begin{align*} EB & = E \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ \mathbf{b}_3^* \end{pmatrix} \\ & \overset{\text{ex}}{=} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -l& 0 & 1 \end{pmatrix} \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ \mathbf{b}_3^* \end{pmatrix} = \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ -l\mathbf{b}_1^* + \mathbf{b}_3^* \end{pmatrix} \end{align*}\] \[\begin{align*} &\implies \text{rows of } AB \text{ is a linear combination of } B \text{'s rows.} \\ &\implies \mathbf{v}^*_i \in C(B^T) \text{, where } \mathbf{v}^*_i \text{ is a row vector of } AB \end{align*}\]
2. Right Multiplication as a Column Operation
\[\begin{align*} AF &= \begin{pmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} F \\ & \overset{\text{ex}}{=} \begin{pmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -l& 0 & 1 \end{pmatrix} = \begin{pmatrix} \mathbf{a}_{1} - l\mathbf{a}_3 & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} \end{align*}\] \[\begin{align*} &\implies \text{columns of } AB \text{ is a linear combination of } A \text{'s columns.} \\ &\implies \mathbf{v}_j \in C(A) \text{, where } \mathbf{v}_j \text{ is a column vector of } AB \end{align*}\]