Cheatsheet: Spans and basis
Core definitions
Section titled “Core definitions”| Term | Definition |
|---|---|
| Linear combination | a · v + b · w for chosen scalars a, b. Scale each vector, add the results. |
| Span | The set of all linear combinations of a set of vectors: every point reachable by adding and scaling. |
| Basis | A linearly independent set that spans the whole space. No redundancy, no gaps. |
| Linearly dependent | At least one vector is in the span of the others (redundant; can be dropped without shrinking the span). |
| Linearly independent | No vector is redundant; each adds a new direction. |
The three span cases (two vectors in 2D)
Section titled “The three span cases (two vectors in 2D)”| Case | Condition | Span |
|---|---|---|
| Generic | Vectors point in different directions | The whole plane (2D) |
| Lined up | One is a scalar multiple of the other | A single line through the origin (1D) |
| Collapsed | Both are the zero vector | Just the origin (a point) |
Two vectors fill the plane exactly when they are linearly independent.
The standard basis
Section titled “The standard basis”i-hat = [1, 0] j-hat = [0, 1]Every 2D vector is a unique linear combination of these:
[3, 4] = 3 · [1, 0] + 4 · [0, 1]The coordinates are the amounts of each basis vector. Uniqueness (exactly one choice of scalars per target) is what lets coordinates be coordinates.
Dimension by stepping up
Section titled “Dimension by stepping up”| Vectors | Largest possible span | Condition to reach it |
|---|---|---|
| One | Line (1D) | Vector is nonzero |
| Two | Plane (2D) | The two are independent (different directions) |
| Three | Space (3D) | None lies in the plane of the other two |
Each independent vector lifts the span by one dimension. The dimension of a space is the size of a basis for it.
Pitfalls to dodge
Section titled “Pitfalls to dodge”- Span is the vectors themselves. No. Span is the whole infinite set of reachable points. Short and long vectors can share a span.
- Span might miss the origin. No.
a = 0, b = 0always reaches the origin. Every span passes through the origin. - Dependent means useless. No. Dependent means redundant for reach, nothing more.
- More vectors means more span. No. Once you span the space, every extra vector is dependent.
Words to use precisely
Section titled “Words to use precisely”- Linear combination: scale-and-add of a set of vectors.
- Span: all reachable points; always a line, plane, (higher flat) through the origin.
- Basis: independent + spanning; the minimal building kit.
- Linearly independent / dependent: whether each vector adds new reach or is redundant.
- Dimension: the number of vectors in a basis.
The one-line version
Section titled “The one-line version”Span is what you can build, a basis is the smallest kit that builds all of it, and dimension is the size of that kit.