> For the complete documentation index, see [llms.txt](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tony-ng-1.gitbook.io/matrices-and-linear-algebra-fundamentals/linear-algebra-pt.-1-2x2.md).

# Linear Algebra (Pt. 1): 2x2

## Geometry of Linear Equation

We will be looking to solve linear algebra for *n* linear equations with *n* unknowns. There are 2 ways to express linear equations in terms of matrices, namely

1. Row Picture
2. Column Picture

### Row Shorthand

We will take reference to the above 2x2 problem, before forward to 3x3 subsequently.

$$
\left\[\begin{array}{cc}
2 & -1\\
-1 & 2
\end{array}\right]
\left\[\begin{array}{cc}
x\\
y
\end{array}\right]
==================

\left\[\begin{array}{cc}
0\\
3
\end{array}\right]
$$

It may also be helpful to lay out the common terminologies used in the video lecture. Dr. Strang would refer to

* **A** as the coefficient matrix
* **x** as the variable matrix, and
* **b** as the constant matrix

![Ax = b](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwOtMcWssLtnH683Or%2F-LwwOzHr0_xhqMrRuK9Q%2Fimage.png?alt=media\&token=e514f104-faf5-4315-8ee6-99350739451e)

As would the name imply, the row picture merely expresses each row of the linear equation into matrices. Thus, there are **no explicit separation** between each variables. Each line of equation still stands as it were, 

* (2x - y = 0) for the first equation
* (-x + 2y = 3) for the second equation

### Row Picture

Hence, if we were to express this in a picture form with these equations, we would merely obtain **lines**. 

![Row Picture of 2x2](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwChi_RDxrucly8e64%2F-LwwF_S8azdz9yxI7obz%2Fimage.png?alt=media\&token=e9712ae3-b726-45d4-9d09-07d411cd2572)

{% hint style="info" %}
Do note an important fundamental concept of:

* **Multiple x and y** values are able to satisfy **each individual equation** (hence we are able to draw a line)
* But only **one x and y value** satisfies both equations simultaneously (the point of intersection 1, 2)
  {% endhint %}

Hence, the solution to the above problem is **x = 1, y = 2**.

### Column Shorthand

Dr. Strang loves his column picture.

In the column picture, there will no longer be an explicit "variable matrix", as **each coefficient matrix** are instead designed to fit for each individual variable. 

In other words, the linear equations are separated out into its **individual columns (variables)** in the form of matrices for the column picture.

$$
x
\left\[\begin{array}{cc}
2 \\
-1
\end{array}\right]
\+
y
\left\[\begin{array}{cc}
-1 \\
2
\end{array}\right]
==================

\left\[\begin{array}{cc}
0\\
3
\end{array}\right]
$$

As opposed to the row picture, there aren't **multiple solutions** of x and y to draw a line for each equation. There only exists **co-ordinates** for each variable, similar to **vectors**. 

### Column Picture

Hence, we can attempt to draw these 3 vectors out to observe its column picture.

![Column Picture](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwQqKo3RAatLo7rkgr%2F-LwwSjJEUh43zAO4iAXD%2Fimage.png?alt=media\&token=4bd96e55-e2db-4c75-8a6e-02f73e749550)

The big question to this would be *"How many combinations of x and y would we require to reach the point of result?"* 

![Linear Combination of the Column Picture](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwUaPej1TGwgBQvPMj%2F-LwwUigHhP4FNEOtnmRF%2Fimage.png?alt=media\&token=1a68b98b-96fb-4f71-9035-43e4fe9e7347)

As you would have known from the Row Picture, the solution is **x = 1, y = 2**. However this time, the solution is obtained through the right combination of vectors. In Mathematics, this is also termed as finding the right **Linear Combination**.

## Side-note

Another interesting way to perform *matrix multiplication* was used as well in the lecture. Personally, I feel this is slightly less intuitive for me, and hence would not put this under the \
Matrix Multiplication" page, but I believe it is good to know as well!

![Column method for Matrix Multiplication](https://3978845189-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LwfxeT-UMPxSGzkwagB%2F-LwwbS5UNujb7r4bkL8k%2F-LwwcyaiER_Zu66yj6Jz%2Fimage.png?alt=media\&token=1236109e-684d-40a5-b1e5-058a1066394d)

#### **More on the next page ⏭**