Linear Algebra II HT23, Matrix multiplication
Flashcards
What are the three views of matrix multiplication?
- Dot product
- Column view
- Row view
$AB = C$. What is $c _ {ij}$ in terms of the rows and columns of $A$ and $B$?
\[c_{ij} = \pmb a^\text{row}_i \cdot \pmb b^\text{col}_j\]
$AB = C$. What is $\pmb c^\text{col} _ j$ in terms of $A$ and $B$?
\[\pmb c_j^\text{col} = A\pmb b_j^\text{col}\]
\[\left(\begin{matrix} a_{11} \& a_{12} \& a_{13} \\\\ a_{21} \& a_{22} \& a_{23} \\\\ a_{31} \& a_{32} \& a_{33} \end{matrix}\right)
\left(\begin{matrix} b_{11} \& b_{12} \\\\ b_{21} \& b_{22} \\\\ b_{31} \& b_{32} \end{matrix}\right)\]
Quick! What’s the 2nd column of the result going to look like?
\[b_{12} \left(\begin{matrix} a_{11} \\\\ a_{21} \\\\ a_{31} \end{matrix}\right) + b_{22}\left(\begin{matrix} a_{12} \\\\ a_{22} \\\\ a_{32} \end{matrix}\right) + b_{32} \left(\begin{matrix} a_{13} \\\\ a_{23} \\\\ a_{33} \end{matrix}\right)\]
$AB = C$. What is $\pmb c _ i^\text{row}$ in terms of $A$ and $B$?
\[\pmb c_i^\text{row} = \pmb a^\text{row}_i B\]
\[\left(\begin{matrix} a_{11} \& a_{12} \& a_{13} \\\\ a_{21} \& a_{22} \& a_{23} \\\\ a_{31} \& a_{32} \& a_{33} \end{matrix}\right)
\left(\begin{matrix} b_{11} \& b_{12} \\\\ b_{21} \& b_{22} \\\\ b_{31} \& b_{32} \end{matrix}\right)\]
Quick! What’s the third row of the result going to look like?
\[a_{31} \left(\begin{matrix} b_{11} \& b_{12} \end{matrix}\right) + a_{32} \left(\begin{matrix} b_{21} \& b_{22} \end{matrix}\right) + a_{33} \left(\begin{matrix} b_{31} \& b_{32} \end{matrix}\right)\]