Introduction

A basis is one of the most important ideas in linear algebra. It gives us a way to describe an entire vector space using only a small, carefully chosen set of vectors. If you already understand linear independence, you’re in a perfect position to learn what a basis is and why it matters.

This section will guide you through:

• What a basis is
• Why bases are useful
• How to check whether a set of vectors forms a basis
• Examples in low‑dimensional spaces
• Exercises and solutions

What Is a Basis?

A basis of a vector space is a set of vectors that satisfies two key properties:

• They are linearly independent.
  No vector in the set can be written as a combination of the others.
• They span the space.
  Every vector in the space can be written as a combination of the vectors in the set.

Put differently:

• A basis is a minimal set of vectors that can generate the entire space.

Some helpful points:

• If you remove any vector from a basis, it will no longer span the space.
• If you add any vector to a basis, the set becomes dependent.
• A basis is like a “coordinate system” for the space.

Examples of Bases

1. Basis of $\mathbb{R}^2$

Common example:

• $(1,0)$
• $(0,1)$

These vectors:

• Are independent
• Span all of $\mathbb{R}^2$

Any vector $(a,b)$ can be written as: $$(a,b) = a(1,0) + b(0,1)$$

2. Another Basis of $\mathbb{R}^2$

The vectors $(1,1)$ and $(1,-1)$ also form a basis.

Why?

• They are independent
• They span the plane

3. Basis of $\mathbb{R}^3$

A standard basis:

• $(1,0,0)$
• $(0,1,0)$
• $(0,0,1)$

But many other sets of three independent vectors also work.

How to Check Whether a Set Is a Basis

To check whether a set of vectors forms a basis:

1. Check independence

• Can any vector be written as a combination of the others?
• If yes → not a basis
• If no → continue

2. Check spanning

• Can every vector in the space be written as a combination of the set?

3. Count the vectors

In $\mathbb{R}^n$:

• Any basis must have exactly $n$ vectors
• If you have fewer → cannot span
• If you have more → must be dependent

This counting rule is extremely powerful.

Why Bases Matter

A basis allows us to:

• Describe vectors using coordinates
• Understand the dimension of a space
• Change between different coordinate systems
• Simplify computations in linear algebra
• Understand geometric transformations

A basis is the “skeleton” of a vector space.

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Exercises

Exercises

1. Determine whether the set $\{(1,2),(3,6)\}$ is a basis for $\mathbb{R}^2$.

   Solution

   The vectors $(1,2)$ and $(3,6)$ are dependent because $(3,6)=3(1,2)$.
   A dependent set cannot be a basis.
   Answer: Not a basis.

2. Does the set $\{(1,0,1),(0,1,1),(1,1,2)\}$ form a basis for $\mathbb{R}^3$?

   Solution

   Check independence:
   Compute determinant of the matrix with these vectors as rows or columns: $$\begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{vmatrix} = 1(1\cdot 2 - 1\cdot 1) - 0 + 1(0\cdot 1 - 1\cdot 1) = 1(1) + 1(-1) = 0$$ Determinant is $0$ → dependent → not a basis.

3. Write the vector $(4,5)$ as a linear combination of the basis $\{(1,1),(1,-1)\}$.

   Solution

   Solve: $$(4,5) = a(1,1) + b(1,-1)$$ This gives:

   • $a + b = 4$
   • $a - b = 5$

   Add equations: $2a = 9 \Rightarrow a = 4.5$
   Then $b = 4 - a = -0.5$

   Answer: $(4,5)=4.5(1,1)-0.5(1,-1)$.

4. True or false: Any set of three independent vectors in $\mathbb{R}^3$ forms a basis.

   Solution

   True.
   In $\mathbb{R}^3$, any set of three independent vectors automatically spans the space.

5. Determine whether the vectors $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$ form a basis for $\mathbb{R}^3$.

   Solution

   Form matrix and compute determinant: $$\begin{vmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{vmatrix} = 1(1\cdot 1 - 0\cdot 1) = 1$$ Determinant is nonzero → independent → basis.

6. Express $(2,3,4)$ in terms of the standard basis of $\mathbb{R}^3$.

   Solution

   In the standard basis: $$(2,3,4)=2(1,0,0)+3(0,1,0)+4(0,0,1)$$

7. Determine whether the set $\{(2,1),(1,2)\}$ spans $\mathbb{R}^2$.

   Solution

   Compute determinant: $$\begin{vmatrix} 2 & 1 \\ 1 & 2 \end{vmatrix} = 4 - 1 = 3$$ Nonzero → independent → spans $\mathbb{R}^2$ → basis.