14. Vector Span

Vector Span

Span is almost the same concept as a subspace — but the two terms carry slightly different emphasis. A subspace is the region of space itself; the span of a set of vectors is how you generate that region through linear combinations.

A subspace is a region of space you can reach by any linear combination of a given set of vectors. Those vectors are said to span that subspace.

1. Formal Definition

Given a set of $n$ vectors $\{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$, their span is the set of all possible linear combinations:

Because the scalars $\alpha_i$ can be arbitrarily large or small, the span always stretches to infinity — just like a subspace.

2. Is a Vector in the Span?

A key question in linear algebra: "Is vector $\mathbf{v}$ inside the span of set $S$?"

• Yes — if $\mathbf{v}$ can be expressed as a linear combination of the vectors in $S$.
• No — if no combination of scalars can produce $\mathbf{v}$ from $S$.

Example

Let the spanning set be: $$S = \left\{ \mathbf{s}_1, \mathbf{s}_2 \right\} = \left\{ \begin{bmatrix}1\\1\\0\end{bmatrix},\; \begin{bmatrix}1\\7\\0\end{bmatrix} \right\}$$

Test two candidate vectors: $$\mathbf{v} = \begin{bmatrix}1\\2\\0\end{bmatrix}, \qquad \mathbf{w} = \begin{bmatrix}3\\2\\1\end{bmatrix}$$

$\mathbf{v} \in \text{span}(S)$? — Yes, using weights $\alpha_1 = \tfrac{5}{6}$ and $\alpha_2 = \tfrac{1}{6}$: $$\frac{5}{6}\begin{bmatrix}1\\1\\0\end{bmatrix} + \frac{1}{6}\begin{bmatrix}1\\7\\0\end{bmatrix} = \begin{bmatrix}1\\2\\0\end{bmatrix} = \mathbf{v} \checkmark$$

$\mathbf{w} \in \text{span}(S)$? — No. Both spanning vectors have a zero third element. No weighted combination of zeros can produce $1$: $$\alpha_1 \cdot 0 + \alpha_2 \cdot 0 = 0 \neq 1 \implies \mathbf{w} \notin \text{span}(S)$$

Finding the correct weights is generally not trivial — it requires matrix computations (solving linear systems). Focus on the concept for now: can the target vector be reached?

3. Geometric Interpretation

One vector → Line

A single non-zero vector spans an infinitely long line through the origin — a 1D subspace.

Two independent vectors → Plane

Two vectors pointing in different directions span an infinite flat plane — a 2D subspace. The 3D plot below illustrates this:

• 🔴 Red — the two spanning vectors $\mathbf{s}_1, \mathbf{s}_2$ (define the plane)
• 🟢 Green — vector $\mathbf{v}$ lies on the plane (in the span)
• 🔵 Blue — vector $\mathbf{w}$ pokes out of the plane (not in the span)

Two independent vectors in $\mathbb{R}^2$ → All of $\mathbb{R}^2$

In 2D space, two independent vectors span the entire plane — every point in $\mathbb{R}^2$ can be reached by some linear combination. The question is only what the weights are.

4. Adding a Redundant Vector Does Not Extend the Span

If you add a third vector already inside the span of the first two, the span does not grow:

Once you span all of $\mathbb{R}^2$, no additional vector can increase it — it is already fully covered.

5. Multiple Sets Can Span the Same Space

Different sets of vectors can span the same subspace. What changes is the coordinate description — i.e., the weights needed to reach any given point.

Example: Same point, two coordinate systems

Using spanning set $S_1 = \{\mathbf{v}_1, \mathbf{v}_2\}$: $$\mathbf{p} = -1 \cdot \mathbf{v}_1 + 0 \cdot \mathbf{v}_2$$

Using a different spanning set $S_2 = \{\mathbf{u}_1, \mathbf{u}_2\}$: $$\mathbf{p} = \frac{3}{2} \cdot \mathbf{u}_1 + \frac{1}{2} \cdot \mathbf{u}_2$$

Same point — but very different coordinates. Some spanning sets are more convenient than others for certain problems. This idea is central to basis, explored in the next lesson.

Key Takeaways

• Span: All possible linear combinations $\alpha_1\mathbf{v}_1 + \cdots + \alpha_n\mathbf{v}_n$ of a given set of vectors.
• Membership: $\mathbf{v} \in \text{span}(S)$ iff $\mathbf{v}$ can be written as a linear combination of the vectors in $S$.
• Geometry: 1 independent vector → line; 2 → plane; $n$ independent vectors in $\mathbb{R}^n$ → all of $\mathbb{R}^n$.
• Redundancy: Adding a dependent vector leaves the span unchanged.
• Multiple spanning sets: Many sets span the same subspace; they differ only in how they coordinatize it.

6. Python: Visualising a Spanning Plane in 3D

# set S
S1 = np.array([1, 1, 0])
S2 = np.array([1, 7, 0])

# vectors v and w
v = np.array([1, 2, 0])
w = np.array([3, 2, 1])

# draw vectors
fig = plt.figure()
# ax = fig.add_subplot(projection='3d')
ax = plt.axes(projection = '3d')
ax.plot([0, S1[0]],[0, S1[1]],[.1, S1[2]+.1],'r',linewidth=3)
ax.plot([0, S2[0]],[0, S2[1]],[.1, S2[2]+.1],'r',linewidth=3)

ax.plot([0, v[0]],[0, v[1]],[.1, v[2]+.1],'g',linewidth=3)
ax.plot([0, w[0]],[0, w[1]],[0, w[2]],'b')

# now draw plane
xx, yy = np.meshgrid(range(-15,16), range(-15,16))
cp = np.cross(S1,S2)
z1 = (-cp[0]*xx - cp[1]*yy)*1./cp[2]
ax.plot_surface(xx,yy,z1)

plt.show()