> [!summary] This note shows an analytical reason why matrix multiplication is done the way it is. >[!info]+ Read Time **⏱ 3 mins** # Motivation > [!warning] Assumptions This motivation for matrix multiplication is not a proof of matrix multiplication, but rather an intuitive idea for why the [[!! Properties of Matrix Multiplication|properties of matrix multiplication]] are true. To create motivation this assume the following: > - A [[Linear Equations|linear equation]] is written as $a_{1}x_{1} + a_{2}x_{2} \dots a_{n}x_{n}=b$ > - The [[Matrix Notation|coefficient]] part of the linear equations can be written as a [[Matrix-Vector Multiplication|matrix-vector multiplication]] > - A [[Vectors Addition & Multiplication|vector]] can have components $\vec{v}=\begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix}$ $ \begin{array}{c} \\ \text{Suppose there are two linear equations} \\ y_{1} = ax_{1}+bx_{2} \\ y_{2} = cx_{1}+dx_{2} \\ \\ \text{Now consider another set of linear equations that are written in terms of the above} \\ z_{1} =Ay_{1} +By_{2} \\ z_{2}= Cy_{1} + Dy_{2} \\ \end{array} $ > [!bug] Goal The goal is to write the $z$ linear equations in terms of the variables $x$. > $ \begin{align*} z_{1 } &= \boxed{\phantom{a}} x_{1}+ \boxed{\phantom{a}}x_{2} \\ z_{2} &= \boxed{\phantom{a}} x_{1}+ \boxed{\phantom{a}}x_{2} \end{align*} > $ Where the boxes are the coefficient the goal of this motivation is to find $ \begin{array}{c} \text{The linear equations $y_{1},y_{2}$ can be subbed into the linear equations $z_1,z_2$} \\ \begin{align*} z_1 &= A(ax_1+bx_2) + B(cx_{1}+dx_{2}) \\ &= (Aa+Bc)x_{1} + (Ab+Bd)x_{2} \\ \\ z_{2}&= C(ax_{1}+bx_{2})+ D(cx_{1}+dx_{2}) \\ &= (Ca + Dc)x_{1} + (Cb+Dd)x_{2} \end{align*} \\ \\ \text{So then this can be written as a matrix vector multipcation}\\ \boxed{\begin{pmatrix} Aa + Bc &Ab+Bd \\ Ca + Dc & Cb + Dd \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix} } \\ \text{Keep note of this} \\ \\ \text{Now, the set of $y$ linear equations can be rewritten as a matrix-vector multiplication} \\ \\ \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} \\ \\ \text{The set of $z$ linear equations can also be rewritten as a matrix-vector multiplication} \\ \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix} \\ \\ \text{So combining these two gives the result} \\ \boxed{\begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix}} \\ \\ \\ \text{If the two boxed equations equal $\vec{z}$, then this comes to a conclusion that}\\ \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} Aa + Bc &Ab+Bd \\ Ca + Dc & Cb + Dd \end{pmatrix} \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/cc1ivDlZ71U?si=-XS42RHGgdXS1wqC" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>