Span

In the last lesson you learned that by choosing different coefficients you can reach different points using a linear combination. Now ask a bigger question: what's every point you can possibly reach?

The span

The span of a set of vectors is the collection of all linear combinations you can make from them — every possible $a\,\mathbf{v}_1 + b\,\mathbf{v}_2$ as $a$ and $b$ range over all real numbers.

It's the answer to: "Where can these vectors take me?"

Two cases

Case 1 — independent vectors

If $\mathbf{v}_1$ and $\mathbf{v}_2$ point in genuinely different directions, their span fills the entire 2D plane. You can reach any point $[x,\ y]$ by picking the right $a$ and $b$. The green shading in the interactive shows this.

Case 2 — parallel vectors

If the two vectors point in the same direction (or exactly opposite), scaling and adding them only ever moves you along that one line. The span collapses from a whole plane down to a single line through the origin.

Why parallel vectors are "redundant"

If $\mathbf{v}_2 = 2\,\mathbf{v}_1$, then $a\,\mathbf{v}_1 + b\,\mathbf{v}_2 = a\,\mathbf{v}_1 + 2b\,\mathbf{v}_1 = (a + 2b)\,\mathbf{v}_1$. The second vector adds no new direction. It can only go where $\mathbf{v}_1$ already goes.

Vectors that are not redundant like this are called linearly independent. That term will come up again and again through the rest of the course.

In higher dimensions

The same idea extends upward. In 3D:

• One vector spans a line
• Two independent vectors span a plane
• Three independent vectors span all of ℝ³

The number of independent vectors you need to span a space is its dimension. 2D space has dimension 2. 3D space has dimension 3.

In code

import numpy as np

v1 = np.array([2, 1])
v2 = np.array([-1, 2])

# Can we reach [3, 4]?
# Solve: a*v1 + b*v2 = [3, 4]
A = np.column_stack([v1, v2])
coeffs = np.linalg.solve(A, [3, 4])
print(coeffs)   # [a, b] — a solution exists because v1, v2 are independent