Introduction

Linear combinations sit at the heart of linear algebra.
If you already know what vectors are, you’re ready to understand how they can be combined to create new vectors.

This idea—simple as it is—powers everything from solving systems of equations to computer graphics and machine learning.

What Is a Linear Combination?

A linear combination of vectors is any expression of the form: $$a_1 v_1 + a_2 v_2 + \dots + a_n v_n$$ where:

• $v_1, v_2, \dots, v_n$ are vectors
• $a_1, a_2, \dots, a_n$ are real numbers (called coefficients)

Key ideas

• You scale each vector by a number.
• Then you add the results.
• The outcome is a new vector.

Example

If

• $v = (1, 2)$
• $w = (3, -1)$

Then a linear combination might be: $$2v - 3w = 2(1,2) - 3(3,-1) = (2,4) - (9,-3) = (-7,7)$$

Why Linear Combinations Matter

Linear combinations allow us to:

• Build new vectors from known ones
• Describe geometric objects (lines, planes, subspaces)
• Understand whether vectors are “enough” to describe a space
• Solve systems of linear equations
• Express concepts like span, basis, and dimension

They are the language of linear algebra.

Geometric Interpretation

Think of vectors as arrows. A linear combination:

• Stretches or shrinks each arrow
• Flips it if the coefficient is negative
• Then adds the arrows tip‑to‑tail

Visual intuition

• All linear combinations of a single non‑zero vector form a line.
• All linear combinations of two non‑parallel vectors in $\mathbb{R}^2$ fill the entire plane.
• In $\mathbb{R}^3$, three non‑coplanar vectors can fill the entire space.

Span

The span of a set of vectors is the collection of all linear combinations of those vectors.

Examples

• The span of $(1,0)$ and $(0,1)$ is the entire plane $\mathbb{R}^2$.
• The span of $(1,2)$ alone is just a line through the origin.
• If two vectors are multiples of each other, their span is still only a line.

Span tells us “how much space” the vectors can reach.

Linear Independence (Briefly)

Vectors are linearly independent if the only way to make the zero vector from a linear combination is by using all zero coefficients.

This matters because:

• Independent vectors give “unique” directions
• They avoid redundancy
• They form the building blocks of a basis

We won’t go deep here, but independence is tightly connected to linear combinations.

Examples

Example 1: A simple combination

Let

• $u = (2,1)$
• $v = (1,-3)$

Compute: $$3u + 2v = 3(2,1) + 2(1,-3) = (6,3) + (2,-6) = (8,-3)$$

Example 2: Describing a line

All vectors of the form: $$t(4, -1)$$ for any real number $t$ form a line through the origin.

Example 3: Describing a plane

All vectors of the form: $$a(1,0,2) + b(0,1,1)$$ fill a plane in $\mathbb{R}^3$.

Exercises

1. Compute the linear combination: $3(1,2) - 2(4,-1)$

   Solution

   Compute $3(1,2) - 2(4,-1)$: $$3(1,2) = (3,6), \quad -2(4,-1) = (-8,2)$$ Adding: $$(3,6) + (-8,2) = (-5,8)$$

2. Let $u = (2,0)$ and $v = (-1,3)$. Compute $5u + v$.

   Solution

   $$5u + v = 5(2,0) + (-1,3) = (10,0) + (-1,3) = (9,3)$$

3. Determine whether the vector $(6,9)$ is a linear combination of $(2,3)$.

   Solution

   Is $(6,9)$ a linear combination of $(2,3)$?

   Yes.
   Since $(6,9) = 3(2,3)$, it is a linear combination.

4. Describe the span of the vector $(5,-5)$ in words.

   Solution

   The span of $(5,-5)$ is a line through the origin in the direction of $(5,-5)$.

5. Are the vectors $(1,2)$ and $(2,4)$ linearly independent?

   Solution

   Are $(1,2)$ and $(2,4)$ linearly independent?

   No.
   $(2,4) = 2(1,2)$, so they are multiples of each other.

6. Write $(3,1)$ as a linear combination of $(1,0)$ and $(0,1)$.

   Solution

   Write $(3,1)$ as a combination of $(1,0)$ and $(0,1)$: $$(3,1) = 3(1,0) + 1(0,1)$$

7. Compute: $-2(3,-1,4) + (1,2,0)$

   Solution

   Compute: $$-2(3,-1,4) + (1,2,0)$$ First: $$-2(3,-1,4) = (-6,2,-8)$$ Then: $$(-6,2,-8) + (1,2,0) = (-5,4,-8)$$

8. True or false: All linear combinations of $(1,1)$ and $(2,2)$ fill the entire plane.

   Solution

   False.
   $(1,1)$ and $(2,2)$ are multiples of each other, so their span is only a line, not the whole plane.