How to Find a Basis Using MATLAB

A basis is a set of linearly independent vectors that spans a space. It's the minimal set needed to describe every vector in that space.

In this guide, we'll use MATLAB to:

1. Extract a basis from a given set of vectors
2. Complete a partial basis to span the entire space

What is a Basis?

A basis for a vector space must satisfy two properties:

1. Linear Independence: No vector in the set can be written as a combination of the others
2. Spans the Space: Every vector in the space can be written as a combination of the basis vectors

For ℝⁿ, a basis must have exactly n vectors.

Step 1: Enter the Vectors

We are given four vectors in ℝ⁴:

v1 = [1 -2 2 -1];
v2 = [-3 5 -2 2];
v3 = [-1 1 2 0];
v4 = [0 -1 4 -1];

Step 2: Construct the Matrix

Place the vectors as columns:

A = [v1.' v2.' v3.' v4.'];

Step 3: Find Independent Vectors

Compute the RREF to identify which vectors are independent:

rref(A)

The pivot columns in the original matrix A correspond to linearly independent vectors.

In this example, suppose columns 1 and 2 have pivots. This means:

• v₁ and v₂ are linearly independent
• v₃ and v₄ can be written as combinations of v₁ and v₂

Step 4: Complete the Basis

Since we only have 2 independent vectors but need 4 for a basis of ℝ⁴, we must add 2 more.

The Identity Matrix Technique

Augment the original matrix with the identity matrix:

I = eye(4);
augA = [A I];
reduced_augA = rref(augA)

This creates an 8-column matrix: [v₁ v₂ v₃ v₄ e₁ e₂ e₃ e₄]

Step 5: Identify the Basis Vectors

After row reduction, the first 4 pivot columns tell us which vectors to keep.

In our example:

• Columns 1 and 2 (v₁ and v₂) have pivots
• Columns 5 and 6 (e₁ and e₂ from the identity) have pivots

Therefore, our basis is:

BasisMatrix = [v1.' v2.' I(:,1) I(:,2)];

This gives us:

BasisMatrix = [v₁ v₂ e₁ e₂]

Why This Works

The identity matrix vectors (e₁, e₂, e₃, e₄) are the standard basis for ℝ⁴. They are guaranteed to be linearly independent.

By augmenting with the identity and row reducing, MATLAB automatically:

1. Keeps independent original vectors
2. Fills in missing dimensions with standard basis vectors

Verification

You can verify your basis by checking:

rank(BasisMatrix)  % Should equal 4

If the rank equals the dimension of the space, you have a valid basis.

Key Takeaways

1. Basis vectors come from pivot columns in the RREF
2. Identity matrix fills missing dimensions when you need to complete a basis
3. Total number of basis vectors = dimension of the space

This technique is essential for understanding subspaces and transformations in linear algebra!