Linear Algebra · step 5 of 13

Basis & dimension

A basis is a linearly independent spanning set, and every basis of a given space has the same number of elements — that number is the dimension. The practical consequence is a shortcut worth remembering: in an n-dimensional space, any n independent vectors automatically span, and any n spanning vectors are automatically independent.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Row reduce to find a basisFor a span, the pivot columns of the original matrix form a basis of the column space; the non-zero rows of the reduced form give a basis of the row space.
Count the dimension as the number of pivotsThat is the rank, and it is the dimension of both the row and the column space.
Extend an independent set to a basisAdjoin standard basis vectors and row reduce; keep the ones that add a new pivot.
Use the shortcutWith the dimension known, checking either independence or spanning suffices — you never need both.

Worked example

Find a basis and the dimension of the subspace of ℝ³ spanned by (1,1,0), (0,1,1) and (1,2,1).

Note (1,1,0) + (0,1,1) = (1,2,1), so the third vector is dependent.
The first two are not multiples of each other, so they are independent.
Hence they span the same subspace as all three.
Two independent vectors spanning the subspace form a basis.

Answer. Basis {(1,1,0), (0,1,1)}, so the subspace has dimension 2 — a plane through the origin in ℝ³.

Where marks get dropped

These are the specific errors that cost credit on basis & dimension questions — QED's rubric penalises each of them separately.

Giving a spanning set that is not independent as a basis. A basis must be both, and redundant vectors must be removed.
Using row-reduced rows as a basis of the COLUMN space. Row reduction preserves the row space but changes the column space — use the original pivot columns.
Assuming the dimension equals the number of vectors given. It equals the number of independent ones.

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Basis & dimension — frequently asked questions

Is a basis unique?

No — every space of dimension ≥ 1 has infinitely many bases. What is unique is the number of elements, which is the content of the dimension theorem.

What is the dimension of the zero subspace?

0, with the empty set as its basis. This is consistent with the rank–nullity theorem.

How do I find a basis for a solution space?

Row reduce, parametrise by the free variables, and the coefficient vectors of the parameters form a basis. The number of free variables is the dimension.

The rest of Linear Algebra

Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.

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