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Matrices as Transformations
Matrix as Transformation
A matrix can represent a rule that transforms an input vector into an output vector.
~/writing/math-foundations-for-reasoning-models $ cat topic.md
Matrix As Transformation
Definition
A matrix can represent a transformation.
That means it can act on an input vector and produce an output vector.
Basic form:
matrix * input vector = output vector
Example idea:
A * v = new v
Matrix as transformation = structured rule for changing vectors.
Transformation From Vector To Vector
A vector can represent a point or data item.
Example:
v = [2, 3]
A matrix can transform it:
v -> transformed v
Example output:
[2, 3] -> [4, 6]
This could represent scaling both components by 2.
A matrix can turn one vector into another vector.
Component Rules
A matrix transformation creates output components from input components.
Example:
input = [x, y]
A transformation might define:
new x = 2*x
new y = 3*y
If:
input = [4, 5]
then:
output = [8, 15]
A matrix packages component update rules.
Transformations Can Scale
A matrix can scale components.
Example rule:
new x = 2*x
new y = 2*y
Input:
[3, 4]
Output:
[6, 8]
The vector keeps its direction but becomes larger.
Scaling is one kind of matrix transformation.
Transformations Can Mix Components
A matrix can also mix components.
Example:
new x = x + y
new y = y
Input:
[3, 4]
Output:
[7, 4]
The first output component uses both input components.
Matrix transformations can combine components, not only scale them separately.
Transformation vs Data Storage
A matrix can be used in two ways:
as data storage
as transformation rule
As data storage:
rows = examples
columns = features
As transformation:
matrix acts on vector
The same rectangular structure can serve different roles.
A matrix's role depends on how it is used.
Why This Matters
Matrix transformations connect algebra, vectors, and computation.
They fit the familiar pattern:
input -> rule -> output
In matrix form:
input vector -> matrix transformation -> output vector
A matrix is a compact way to describe structured vector computation.
Final Summary
- A matrix can represent a transformation.
- It can map an input vector to an output vector.
- It packages component update rules.
- It can scale, mix, or rearrange components.
- A matrix can also be used as data storage.
- Matrix transformations are structured input-output rules.
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Continue the series.
Matrix-Vector Multiplication