Matrix-Vector Multiplication

matrix-vector

We will denote ℝ^n as a set.

$\begin{array}{r}{\mathbb{R}}^n=(r_1,r_2...,r_n):v_i\in \mathbb{R}\end{array}$

Choose the elements as vectors (or n-vectors):

$\left[\begin{array}{ccc}{r}_1& r_2& r_n\end{array}\right]$ or $\left[\begin{array}{c}{r}_1\\ r_2\\ r_n\end{array}\right]$

the multiplication part

If A is an m*n matrix, it is often convenient to view it as a row of columns. That is, if $a_1,a_2,...$ are the columns, we write: $\begin{array}{r}A=\left[\begin{array}{ccc}{a}_1& a_2& a_n\end{array}\right]\end{array}$

Consider any system of linear equations with m*n coefficient matrix A. If b is the constant matrix of the system and $x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$, the system can be written as the single vector equation: $x_1{a}_1{x}_2{a}_2...+x_n{a}_n=b$.

So given that, we can structure this as:

$\begin{array}{r}A=a_1{x}_1{a}_2{x}_2...+a_n{x}_n=\left[\begin{array}{c}{b}_1\\ b_2\\ ...\\ b_m\end{array}\right]\end{array}$

a_1 is a full column of things in the broader m*n matrix.

So… tl;dr turn the matrix into vectors, sub in a variable for the vector, then it simplifies into an equation you can do something about. (I think)

Example

Write the system

$\{\begin{array}{l}3x_1+2x_2-4x_3=0\\ x_1-3x_2+x_3=3\\ x_2-5x_3=-1\end{array}$

in the form $x_1{a}_1{x}_2{a}_2...+x_n{a}_n=b$.

$\begin{array}{r}{x}_1\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right]+x_2\left[\begin{array}{c}2\\ -3\\ 1\end{array}\right]+x_3\left[\begin{array}{c}-4\\ 1\\ -5\end{array}\right]=\left[\begin{array}{c}0\\ 3\\ -1\end{array}\right]\end{array}$

Def: Marix vector multiplication :drill: Let A be given in form $A=[a_1,a_2,...,a_n]$ be an m*n matrix, written in terms of its columns.

If x is a vector given by $X=\left[\begin{array}{c}{x}_1\\ x_2\\ ...\\ x_n\end{array}\right]$ is any n-vector, then the product:

$\begin{array}{r}Ax=\left[\begin{array}{cccc}{a}_1& a_2& ...& a_n\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ...\\ x_n\end{array}\right]\end{array}$ is defined by [ $Ax=x_1{a}_1{x}_2{a}_2...+x_n{a}_n$ ]

observations

1. Every system of linear equations has the form $Ax=b$ where A is the coefficient matrix, b is the constant matrix and x is the matrix of variables.
2. The system $Ax=b$ is consistent IFF b is a linear combination of the columns of A
3. $a_1,a_2,...,a_n$ are the columns of A and if x is the vector

$\left[\begin{array}{c}{x}_1\\ x_2\\ ...\\ x_n\end{array}\right]$

, then x is the solution to the linear system Ax=b IFF $x_1,x_2,...,x_n$ are a solution of the vector equation $x_1{a}_1{x}_2{a}_2...+x_n{a}_n=b$.

Examples $$ \begin{aligned} A = \begin{bmatrix} 2&-1&3&5\ 0&2&-3&1\ -3&4&1&2 \end{bmatrix}\

X = \begin{bmatrix} 2\ 1\ 0\ -2 \end{bmatrix}\

AX = \begin{bmatrix} 2&-1&3&5\ 0&2&-3&1\ -3&4&1&2 \end{bmatrix}\begin{bmatrix} 2\ 1\ 0\ -2 \end{bmatrix}

= 2 \begin{bmatrix} 2\ 0\ -3 \end{bmatrix} + 1 \begin{bmatrix} -1\ 2\ 4 \end{bmatrix} + 0 \begin{bmatrix} 3\ -3\ 1 \end{bmatrix} - 2 \begin{bmatrix} 5\ 1\ 2 \end{bmatrix}

= \begin{bmatrix} -7 \ 0 \ -6 \end{bmatrix} \end{aligned} $$

Remark :drill: Scheduled: 2022-10-22 When an mn matrix A is multipled with a column vector v, the definition requires that v must be of size [n1]

Solve the following system x_1 - x_2 - x_3 + 3x_4 = 2 2x_1 - x_2 - 3x_3 + 4x_4 = 6 x_1 - 2x_3 + x_4 = 4

Augmented matrix:

$\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ 2& -1& -3& 4& |& 6\\ 1& 0& -2& 1& |& 4\end{array}\right]$

r2 = r2-r1*2 $\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ 0& 1& -1& -2& |& 2\\ 1& 0& -2& 1& |& 4\end{array}\right]$

r3 = r3 - r1 $\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ 0& 1& -1& -2& |& 2\\ 0& 1& -1& -2& |& 2\end{array}\right]$

r3 -= r2 $\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ 0& 1& -1& -2& |& 2\\ 0& 0& 0& 0& |& 0\end{array}\right]$

r1 += r2 $\left[\begin{array}{cccccc}1& 0& -2& 1& |& 4\\ 0& 1& -1& -2& |& 2\\ 0& 0& 0& 0& |& 0\end{array}\right]$

$\begin{array}{rl}{x}_1-2s+t& =4\\ x_1& =4+2s-t\\ x_2-s-2t& =2\\ x_2& =2s2t\\ x_3& =s\\ x_4& =t\end{array}$

$\left[\begin{array}{c}4+2s-t\\ 2+s+2t\\ s\\ t\end{array}\right]$

= \begin{bmatrix} 4\\ 2\\ 0\\ 0\\ \end{bmatrix} + s \begin{bmatrix} 2\\ 1\\ 1\\ 0 \end{bmatrix} + t \begin{bmatrix} -1\\ 2\\ 0\\ 1 \end{bmatrix} \end{aligned} $$ (we get that 4 2 0 0 matrix by setting the parameters to zero) 2,1,1,0 & -1,2,0,1 are the solutions to associated [[Homogeneous Equations|Homogeneous system]]. (determined by setting all the parameters to zero rather than 2,6,4 that we used above) ### Notes Wen a solution to the system AX=b exists, the solution be the sum of a particular solution obtained by setting the parameters to be [0] plus the solution of the [associated homogenous system|what kind of system]. ### def :drill: The [identity matrix] is the matrix $$ \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} $$ or $$ \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} $$ ### Matrix transformation Consider the transformation of $\mathbb{R}^2$ given by the reflection in the x-axis, [a_1, a_2] turns into [a_1, -a_2]. Apparently x/y coords are written as $$ \begin{bmatrix} a_1\\ a_2 \end{bmatrix} $$ not $$ \begin{bmatrix} a_1&a_2 \end{bmatrix} $$ $$ \begin{aligned} \begin{bmatrix} a_1\\ a_2 \end{bmatrix} = \begin{bmatrix} a_1\\ 0 \end{bmatrix} + \begin{bmatrix} 0\\ -a_2 \end{bmatrix} \\ = a_1 \begin{bmatrix} 1\\ 0 \end{bmatrix} + a_2 \begin{bmatrix} 0\\ -1 \end{bmatrix} \\ = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} a_1\\ a_2 \end{bmatrix} \end{aligned} $$ So reflecting $$ \begin{bmatrix} a_1\\ a_2 \end{bmatrix} $$ in the x-axis can be achieved by multiplying by the matrix $$ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} $$ Thus, the reflection is a function $T_A: \mathbb{R}^2 -> \mathbb{R}^2$ given by $T_Ax = Ax$ for all x in $\mathbb{R}^2$ where: $$ A = \begin{smallmatrix} 1&0\\ 0&-1 \end{smallmatrix} $$ ### def :drill: Scheduled: 2022-10-22 $T_A$ is called the [matrix transformation] induced by A. ### note In general, if A is an m*n matrix, multiplication by A gives a transformation $T_A: \mathbb{R}^n -> \mathbb{R}^m$ defined by $T_A(x) = Ax$ for every x in $\mathbb{R}^n$. This goes from n -> m because the x in $Ax$ is [an $n \cdot 1$ matrix|shape].