> [!summary] This note simplifies a system of linear equations by using matrices and vector manipulation. > **Key Result:** $A \vec{x} = x_{1} \vec{a_{1}} + x_{2} \vec{a_{2}}+ \dots + x_{n} \vec{a_{n}} = \vec{b}$ >[!info]+ Read Time **⏱ 3 mins** # Definition Matrix-vector multiplication manipulation of a system of [[Linear Equations|linear equations]] into a [[Matrix Notation|coefficient matrix]] and the associated [[Scalar & Vectors|vector]]. ## Derivation > [!warning] Assumptions To derive an equation for matrix-vector multiplication, assume the following: > - A [[Linear Equations|system of linear equations]], with $m$ equations and $n$ variables, looks like: > $ \begin{align*} a_{1}x_{1} + a_{2}x_{2} \dots a_{n}x_{n} &=b_{1} \\ a_{21}x_{1} + a_{22}x_{2} \dots a_{2n}x_{n}&=b_{2} \\ \vdots\\ a_{m1}x_{1} + a_{m2}x_{2} \dots a_{mn}x_{n}&=b_{m} \end{align*} > $ > - The [[Matrix Notation|coefficient matrix]] is denoted as: > $ A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} > $ > - The [[Row & Columns Vectors|column vector]] is denoted as: > $c_{n} = \begin{bmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{bmatrix} > $ > - From the properties of [[!! Properties of Matrix Addition & Scalar Multiplication|matrix addition and multiplication]] two matrices can be added together to give the result $\begin{bmatrix} a \\ b\end{bmatrix}+ \begin{bmatrix} c \\ d \end{bmatrix}=\begin{bmatrix}ac \\ bd\end{bmatrix}$ Given a system of linear equation the linear equations can be rewritten using the column vector to denote the constants ($a_{mn}$) and the solution values $b_{m}$ to give the following result. $ x_1 \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix} + x_2 \begin{bmatrix} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{bmatrix} + \cdots + x_n \begin{bmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix} $ The constant column vectors are in $\mathbb{R}^m$. The unknown variables ($x_{1}+\dots+x_{n}$) are in $\mathbb{R}^n$. The resultant variables ($b_{1}\dots b_{m}$) is in $\mathbb{R}^m$. Combining the constant columns vector and multiplying it with unknown variables gives the following result. > [!note] The variables ($x_{1}+\dots+x_{n}$) are written as a matrix with $n$ columns or $\begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n}\end{bmatrix}$ $ \underbrace{ \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} }_{ A } \underbrace{ \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} }_{ \vec{x} } = \underbrace{ \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix} }_{ \vec{b} } $ > [!bug] Simplification This is often simplified in the following ways: > - Letting the whole constant matrix be written as $A$ > - Letting each column vector and solutions values be written as vector (e.g. $\begin{bmatrix}a_{11} \\a_{21} \\\vdots \\a_{m1}\end{bmatrix}=\vec{a_{1}}$) > - The variables and constant vectors can be written as [[Vector Equations|a vector equation]] This simplification results in: $ A \vec{x} = x_{1} \vec{a_{1}} + x_{2} \vec{a_{2}}+ \dots + x_{n} \vec{a_{n}} = \vec{b} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/4uJNuGfbxLg?si=wSlSdGZNLsTeFIM7" 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>