Introduction

The idea of span answers a simple but powerful question:

Given some vectors, what other vectors can we reach using linear combinations of them?

If you already understand linear combinations, span is the natural next step. It tells us the shape or space formed by all possible combinations.

What Is Span?

The span of a set of vectors is: $$\text{Span}(v_1, v_2, \dots, v_n) = \{a_1 v_1 + a_2 v_2 + \dots + a_n v_n \mid a_i \in \mathbb{R}\}$$

Key points

• You take all possible linear combinations.
• The result is a collection of vectors, not just one.
• The span describes the space those vectors can reach.

Intuition

• One vector spans a line.
• Two non-parallel vectors in $\mathbb{R}^2$ span the entire plane.
• Two parallel vectors still only span a line.

Geometric View

Span is easiest to understand visually.

If you have:

• One vector in the plane → you get a line through the origin.
• Two non-parallel vectors → you get the whole plane.
• Three vectors in $\mathbb{R}^3$ that don’t lie in the same plane → you get all of 3‑space.

Why geometry helps

• Span tells you the “reach” of your vectors.
• It shows whether your vectors cover a line, a plane, or something larger.

Examples

Example 1: Span of a single vector

Let $v = (3,1)$.
Then: $$\text{Span}(v) = \{ t(3,1) \mid t \in \mathbb{R} \}$$ This is a line through the origin.

Example 2: Span of two vectors in the plane

Let

• $u = (1,0)$
• $v = (1,1)$

Because these are not multiples of each other, their span is the entire plane $\mathbb{R}^2$.

Example 3: Span of dependent vectors

Let

• $u = (2,4)$
• $v = (1,2)$

Since $u = 2v$, they point in the same direction.
Their span is just a line, not a plane.

Example 4: Span in 3D

Let

• $u = (1,0,0)$
• $v = (0,1,0)$

Then: $$\text{Span}(u,v) = \{(a,b,0) \mid a,b \in \mathbb{R}\}$$ This is a plane inside $\mathbb{R}^3$.

Why Span Matters

Span is foundational because it helps us understand:

• Whether a set of vectors can describe an entire space
• Whether we have “enough” vectors to solve a system
• How to build concepts like basis and dimension
• Whether vectors are redundant or useful

Span is one of the simplest and most important ideas in linear algebra.

Exercises

1. Describe the span of the vector $(4,2)$ in words.

   Solution

   The span of $(4,2)$ is a line through the origin in the direction of $(4,2)$.

2. Determine whether the vectors $(1,2)$ and $(3,6)$ span a line or the entire plane.

   Solution

   $(3,6)$ is a multiple of $(1,2)$, so the vectors are dependent.
   They span only a line, not the plane.

3. Compute a general expression for all vectors in $\text{Span}((1,0,1),(0,1,1))$.

   Solution

   A general vector in the span is: $$a(1,0,1) + b(0,1,1) = (a, b, a+b)$$

4. True or false: The vectors $(1,0)$ and $(0,0)$ span the entire plane.

   Solution

   False.
   $(0,0)$ contributes nothing, so the span is just the line generated by $(1,0)$.

5. Let $u = (2,1)$ and $v = (1,-1)$. Do $u$ and $v$ span the plane?

   Solution

   The vectors $(2,1)$ and $(1,-1)$ are not multiples of each other.
   Therefore, they span the entire plane.

6. Write a vector in $\text{Span}((1,2),(2,1))$ by choosing your own coefficients.

   Solution

   Example answer: $$3(1,2) - 2(2,1) = (3,6) - (4,2) = (-1,4)$$ Any linear combination is acceptable.

7. Describe the span of $(0,0,1)$ in $\mathbb{R}^3$.

   Solution

   The span of $(0,0,1)$ is a line along the $z$‑axis.

8. Determine whether $(3,1)$ is in the span of $(1,0)$ and $(0,1)$.

   Solution

   Yes.
   $(3,1) = 3(1,0) + 1(0,1)$, so it is in the span.