Span

Definition

Span is the set of all vectors that can be built from given vectors using linear combinations.

Example:

v = [1, 0]
w = [0, 1]

Using:

a * v + b * w

we can build:

[a, b]

So these two vectors span all two-component vectors.

Span = everything you can build from available vectors.

---

Span With One Vector

Start with one vector:

v = [1, 2]

All scalar multiples are in its span:

0 * v = [0, 0]
1 * v = [1, 2]
2 * v = [2, 4]
-1 * v = [-1, -2]

This creates all vectors in that same direction.

One vector spans all scaled versions of itself.

---

Span With Two Independent Vectors

Use:

[1, 0]
[0, 1]

Linear combinations create:

a * [1, 0] + b * [0, 1] = [a, b]

By choosing different a and b, we can create any two-component vector.

Independent vectors can span a larger space.

---

Span With Redundant Vectors

Use:

[1, 0]
[2, 0]

Both point in the same direction.

Any linear combination gives:

a * [1, 0] + b * [2, 0] = [a + 2b, 0]

The second component is always 0.

So these vectors cannot build [0, 1].

Redundant vectors limit the span.

---

Span As Reachability

Span answers a reachability question:

Which vectors can be reached using these building blocks?

Example:

starting vectors: [1, 0], [0, 1]
reachable: [3, 5]

because:

3 * [1, 0] + 5 * [0, 1] = [3, 5]

Span describes what can be constructed.

---

Span And Basis

A basis is a set of independent vectors that spans a space.

Example:

[1, 0]
[0, 1]

These vectors are:

independent
able to build all two-component vectors

So they form a basis for two-dimensional vectors.

Basis = independent vectors that span the space.

---

Why Span Matters For Data

If data lives in a vector structure, span tells what kinds of data patterns can be represented.

If the available vectors span only a limited set, then some vectors cannot be built.

That means the representation has limits.

Span connects available structure to representable possibilities.

---

Final Summary

• Span is the set of vectors buildable from given vectors.
• It uses linear combinations.
• One vector spans scaled versions of itself.
• Independent vectors can span more structure.
• Redundant vectors limit what can be built.
• A basis is independent and spans the space.
• Span describes representational reach.