# Topic 1: Systems of Equations, Gaussian Elimination

###### tags: `345`

## Quiz 1 (Thursday 09/13)

:::info
[Previous Quiz 1 with solutions for study.](https://ketchers.github.io/Teaching/345/Topic%201/Quiz_1_Ground__091622_soln.pdf)
:::warning
[Solutions](https://ketchers.github.io/Teaching/345/Topic%201/Quiz1_soln.pdf) For now I have not included the *reasoning* for the T/F. This is in case I decide to let you improve your score by submitting your own reasoning.
:::

## Homework 1 (Due Sunday 09/15)

:::info
* 1.1: 8, 9 (a-c)
* 1.2: 5 (c, d, e) , 8, 11, 22 (b,c)
* 1.3: 6, 7, 13, 16
* 1.4: 8, 10, 22, 23, 27, 30, 33
* 1.5: 8 (b,d), 19, 28, 32

:::warning
[Solutions](https://ketchers.github.io/Teaching/345/Topic%201/Homework_1_Partial_Solutions.pdf)
:::

### Brief intro to vectors and their operations.

Here is a [good video](https://youtu.be/fNk_zzaMoSs) introduction from the [Linear Algebra series](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) at [3Blue1Brown](https://www.youtube.com/@3blue1brown).

Here is a [Geogebra app](https://www.geogebra.org/calculator/wadahkuc) for playing with scalar multiplication and addition for vectors in 2D. Here is a similar [Geogebra app](https://www.geogebra.org/calculator/nadgujjn) for playing with the same operations in 3d.

### Quick introduction to matrix and vector notation

An ***$m\times n$-matrix*** is just a 2-dimensional array of numbers (***scalars***) with $m$ rows and $n$ columns. The $(i,j)$ entry of an $m\times n$-matrix $A$ will be denoted $A_{i,j}$ or sometimes $a_{i,j}$ (context).

We often view matrices via columns or rows.

**Example**
$$
\begin{bmatrix}
1&2&3\\4&5&6
\end{bmatrix}=
\begin{bmatrix}
\begin{bmatrix}1\\4\end{bmatrix}&
\begin{bmatrix}2\\5\end{bmatrix}&
\begin{bmatrix}3\\6\end{bmatrix}
\end{bmatrix}=
\begin{bmatrix}
\begin{bmatrix}1&2&3\end{bmatrix}\\
\begin{bmatrix}4&5&6\end{bmatrix}
\end{bmatrix}
$$
We will officially say that vectors in $\mathbb R^n$ are $n\times 1$ matrices (column vectors). So an $m\times n$-matrix can be thought of as $n$ (column vectors) from $\mathbb R^m$. The rows might also naturally be considered as vectors (row vectors), so an $m\times n$ matrix might also be viewed as $m$ row vectors from $\mathbb R^n$.

We won't use this often, but the $i$^th^ row vector of $A$ might be denoted $A_{i,*}$ while the $j$^th^ column vector would be $A_{*,j}$.

We will define matrix multiplication in a number of ways by dividing one or both matrices into columns or rows and each has a different intuitive interpretation. One we need now is the product of a matrix and a vector:
$$
\begin{align}
A\mathbf x
&=\begin{bmatrix}\mathbf a_1&\mathbf a_2&\cdots&\mathbf a_n\end{bmatrix}
\begin{bmatrix}x_1\\x_2\\\vdots\\ x_n\end{bmatrix}\\
&=x_1\mathbf a_1+x_2\mathbf a_2+\cdots+x_n\mathbf a_n=\sum_{i=1}^nx_i\mathbf a_i\in\mathbb R^m
\end{align}
$$

**Example**
$$
\begin{align}
\begin{bmatrix}
1&2&3\\4&5&6
\end{bmatrix}
\begin{bmatrix}7\\8\\9\end{bmatrix}&=
\begin{bmatrix}
\begin{bmatrix}1\\4\end{bmatrix}&
\begin{bmatrix}2\\5\end{bmatrix}&
\begin{bmatrix}3\\6\end{bmatrix}
\end{bmatrix}\begin{bmatrix}7\\8\\9\end{bmatrix}\\
&=7\begin{bmatrix}1\\4\end{bmatrix}+
8\begin{bmatrix}2\\5\end{bmatrix}+
9\begin{bmatrix}3\\6\end{bmatrix}\\
&=\begin{bmatrix}7\\28\end{bmatrix}+
\begin{bmatrix}16\\40\end{bmatrix}+
\begin{bmatrix}27\\54\end{bmatrix}\\
&=\begin{bmatrix}50\\122\end{bmatrix}
\end{align}
$$

We now apply this to equations of the form $A\mathbf x=\mathbf b$.

### Two ways to interpret/visualize solving systems.

Here is a [great video from Zach Star](https://youtu.be/4csuTO7UTMo) showing the two different geometric interpretations of $Ax=b$.

I have made two Geogebra apps that represent the two points of view:

The system I consider is:
$$
\begin{bmatrix}
3&2&1\\4&-2&-5\\1&2&-3
\end{bmatrix}
\begin{bmatrix}
x\\y\\z
\end{bmatrix}=
\begin{bmatrix}
6\\-3\\0
\end{bmatrix}
$$

* [Solving the vector equation](https://www.geogebra.org/3d/nadgujjn)
$$
\begin{bmatrix}
3\\4\\1
\end{bmatrix}x+
\begin{bmatrix}
2\\-2\\2
\end{bmatrix}y+
\begin{bmatrix}
1\\-5\\-3
\end{bmatrix}z=
\begin{bmatrix}
6\\-3\\0
\end{bmatrix}
$$

* [Solving for the intersection of planes](https://www.geogebra.org/3d/hyh58anv)
$$
\begin{align}
3x+2y+\hphantom{5}z&=6\\
4x-2y-5z&=-3\\
x+2y-3z&=0
\end{align}
$$

[Here is a 2D version](https://www.geogebra.org/calculator/dpejuvez) where both interpretations are presented together. The system here is
$$
\begin{bmatrix}
3&-2\\-2&-1
\end{bmatrix}\begin{bmatrix}
x\\y
\end{bmatrix}=\begin{bmatrix}
b_0\\b_1
\end{bmatrix}
$$
or
$$
\begin{bmatrix}
3\\-2
\end{bmatrix}x+\begin{bmatrix}
-2\\-1
\end{bmatrix}y=\begin{bmatrix}
b_0\\b_1
\end{bmatrix}
$$
where you get to play with $(b_0,b_1)$.

## [LU and CR decomposition](/xhhDq_45S4aBJ2AmeKfn4Q?view)

## [Matrix Algebra and *LU* decomposition](/j9l_uh9vT0a_lMSAt6HQRA)

[TBA]:
/DCvTLDbxRBKXK4R5QWsEtQ?view "TBA"