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The Basis Theorem

The Basis Theorem is a powerful shortcut: for an $n$ -dimensional subspace, you don't need to verify both spanning and independence separately. If a set of exactly $n$ vectors satisfies either condition, it automatically satisfies both.

This means: to show a set of $n$ vectors is a basis for an $n$ -dimensional space, just show it spans OR just show it's independent — the other condition is free.

In practice, this often halves the work of basis verification. For $\mathbb{R}^n$ : $n$ vectors that span $\mathbb{R}^n$ are automatically independent; $n$ independent vectors in $\mathbb{R}^n$ automatically span $\mathbb{R}^n$ .

Formal View

Theorem 2.15 (Basis Theorem)

Let

V

be a subspace of dimension

n

. A set

\{\mathbf{a}_1, \ldots, \mathbf{a}_n\}

of exactly

n

vectors in

V

is a basis for

V

if and only if it spans

V

, and also if and only if it is linearly independent.

Why This Matters

The Basis Theorem is why proving a set is a basis only requires checking one of the two conditions when the count is right.

Verifying a set of $n$ orthonormal vectors is a basis for $\mathbb{R}^n$ — just check orthonormality
In coding theory, checking that a code has the right dimension verifies both independence and spanning

Learning Resources

Basis theorem and change of basis

MIT OpenCourseWare

Proof and applications of the basis theorem.

49 min

Quiz

Question 1

Three linearly independent vectors in $\mathbb{R}^3$ automatically form a basis for $\mathbb{R}^3$ .

Common Mistakes

Applying the Basis Theorem without confirming the count: you must have exactly $n$ vectors for an $n$ -dimensional space.
Thinking you always need to verify both spanning and independence independently.