Computer Mathematics Program to Demonstrate Matrices and Linear Algebra (Computer-Oriented) — the math behind images, graphics, and ML
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All about Computer Mathematics - Matrices and Linear Algebra (Computer-Oriented) — the math behind images, graphics, and ML
Jan 27, 2026
## Chapter 9: Matrices and Linear Algebra (Computer-Oriented) — the math behind images, graphics, and ML
I’m going to start with a simple question that makes students pause:
If a computer can store a photo, what is that photo *as numbers*?
Because a computer does not store “a cat” or “a selfie”.
It stores a big table of values.
That table is a matrix.
And once you accept “a thing can be a table of numbers”, a lot of computer topics stop feeling magical:
- images (pixels in a grid)
- game graphics (moving and rotating objects)
- machine learning (data as rows and columns)
- linear systems (solving many equations together)
### Why this chapter exists (the real reason)
Matrices often show up as a scary school topic:
“2×3, 3×2, multiply, determinant…”
But in programming, matrices are not scary. They’re just a clean way to represent:
- many numbers at once
- many relationships at once
- many transformations at once
It’s the “batch” version of normal arithmetic.
### The biggest confusion: matrix multiplication feels random
People ask:
“Why isn’t matrix multiplication like normal multiplication?”
Because it’s not made to be “number × number”.
It’s made to represent a *function* that transforms vectors.
This is the mindset:
- vector = a point or a list of features
- matrix = a machine that changes that vector
So matrix multiplication is designed to compose machines.
#### Basic: an image is a matrix (grayscale)
Take a tiny 3×3 grayscale image. Each pixel is 0 to 255.
```text
[
[ 0, 50, 255],
[ 30, 80, 200],
[ 10, 10, 10]
]
```
This is literally a matrix: rows and columns.
- 0 = black
- 255 = white
- numbers in between = gray
Now you can do “math” on the image:
- brighten: add a number to every cell
- darken: subtract
- invert: replace x with (255 - x)
You didn’t need any “photo magic”. Just matrix operations.
#### Common mistake: mixing up “rows” and “columns” (and getting rotated data)
This happens in real code.
You have data shaped like:
- rows = users
- columns = features (age, height, score…)
But you accidentally treat columns as users.
Then your model learns nonsense. Or your output looks rotated.
Quick tip:
Say out loud:
“One row = one item.”
If you can’t say that sentence, you don’t know your matrix meaning yet.
This mistake is common in:
- CSV handling
- pandas / numpy style code
- ML pipelines
And the bug is painful because the program still runs.
#### Simple reason: dot product (why it shows up everywhere)
Dot product is the simplest “vector math” that actually matters.
You have two lists:
```text
a = [a1, a2, a3]
b = [b1, b2, b3]
```
Dot product is:
```text
a · b = a1*b1 + a2*b2 + a3*b3
```
Why programmers should care:
- it measures “matching” between two vectors
- it’s the core of matrix multiplication
- it shows up in recommendation, search, ML, graphics lighting
Even if you forget the formula, remember the meaning:
“Multiply matching positions, then add.”
#### Practical: matrix multiplication is “many dot products”
Let’s keep it tiny and readable.
Matrix (2×3):
```text
M =
[
[1, 2, 3],
[4, 5, 6]
]
```
Vector (3×1):
```text
v =
[
[10],
[20],
[30]
]
```
Now M × v gives a (2×1) result.
Row 1 result:
```text
1*10 + 2*20 + 3*30 = 140
```
Row 2 result:
```text
4*10 + 5*20 + 6*30 = 320
```
So:
```text
M*v =
[
[140],
[320]
]
```
This is why shapes matter:
- inner sizes must match (3 with 3)
- output size is (rows of M) × (cols of v)
It’s easy to try to multiply any two matrices and get stuck.
But this is not random rule. It is a compatibility rule: “can this machine accept that input?”
#### Practical + CS-useful: identity matrix (the “do nothing” operation)
Identity matrix is the matrix version of number 1.
For 3D features it looks like:
```text
I =
[
[1,0,0],
[0,1,0],
[0,0,1]
]
```
When you multiply I with a vector, the vector stays same.
Why it matters in real work:
- it is used in transformations as “no change”
- it helps in debugging pipelines (insert identity to check where change happens)
- it shows up in solving systems and in ML math
If you ever see “initialize weights like identity”, that is the idea: start with no weird change.
#### Advanced but simple: transformations in graphics (scale + rotate idea)
Graphics people love matrices because one matrix can represent a transformation.
In 2D, a point is a vector:
```text
p = [x, y]
```
A scaling transformation can be represented as:
```text
S =
[
[sx, 0 ],
[0 , sy]
]
```
Then:
```text
S * p = [sx*x, sy*y]
```
That’s “stretch x and y differently”.
Rotation is also a matrix (it uses sin/cos).
You don’t need to memorize the exact rotation matrix now.
Just keep the pro idea:
Matrix = transformation machine
Multiplying matrices = combining transformations
So in games and graphics pipelines, you see chains like:
```text
final = Projection * View * Model * point
```
That’s not fancy math showing off. That’s just “apply these transformations in order”.
### One honest warning (so you don’t get fooled)
Linear algebra becomes heavy when people jump straight into:
- determinants
- eigenvalues
- proofs
We’ll reach those slowly later. For now you only need:
- data as tables (matrices)
- transformations as matrices
- dot product and multiplication as “how machines work”
That’s enough to understand 80% of practical use in CS.
---
## Conclusion
In this article, we explored the core concepts of All about Computer Mathematics - Matrices and Linear Algebra (Computer-Oriented) — the math behind images, graphics, and ML. Understanding these fundamentals is crucial for any developer looking to master this topic.
## Frequently Asked Questions (FAQs)
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A: All about Computer Mathematics - Matrices and Linear Algebra (Computer-Oriented) — the math behind images, graphics, and ML is a fundamental concept in this programming language/topic that allows developers to perform specific tasks efficiently.
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