~/writing/math-foundations-for-reasoning-models $ ls
Vectors as Data
Linear Independence
Linear independence means vectors provide directions that cannot be built from each other.
~/writing/math-foundations-for-reasoning-models $ cat topic.md
Linear Independence
Definition
Vectors are linearly independent when none of them can be built from the others using a linear combination.
Simple example:
[1, 0]
[0, 1]
Neither vector can be made by scaling the other.
They provide different directions.
Linear independence = no vector is redundant.
Dependent Vectors
Vectors are linearly dependent when at least one vector can be built from the others.
Example:
[1, 0]
[2, 0]
The second vector is:
2 * [1, 0] = [2, 0]
So it does not add a new direction.
Linear dependence means at least one vector repeats existing structure.
Independent Directions
In two dimensions:
[1, 0]
[0, 1]
give two independent directions.
One controls the first component.
One controls the second component.
Together they can build any vector like:
[3, 5]
using:
3 * [1, 0] + 5 * [0, 1]
Independent vectors expand what can be built.
Redundancy
Linear dependence means redundancy.
Example:
v = [1, 2]
w = [2, 4]
Since:
w = 2 * v
w does not add a new direction.
It gives more of the same direction.
Dependent vectors add size, not new structure.
Why Independence Matters
Independence tells us whether a set of vectors carries unique directions.
This matters when building a coordinate system.
A useful basis should avoid redundant vectors.
Example:
[1, 0]
[0, 1]
is useful in two dimensions because each vector contributes something the other cannot.
Independence is needed for clean vector representation.
Not About Meaning
Linear independence is structural.
It does not depend on what the vectors mean in the real world.
Example:
[1, 0]
[0, 1]
They may represent positions, measurements, or abstract data components.
The independence relation is about whether one can be built from the other.
Linear independence is about construction, not interpretation.
Final Summary
- Linearly independent vectors cannot be built from each other.
- Linearly dependent vectors contain redundancy.
- Scaling one vector cannot create a new independent direction.
- Independent vectors add new structure.
- Basis vectors should be independent.
- Independence is structural, not semantic.
~/writing/math-foundations-for-reasoning-models $ open
Continue the series.
Span