Linear Algebra · step 9 of 13

Column space, null space & solution sets

The column space is the span of the columns — exactly the set of b for which Ax = b is solvable. The null space is the set of x with Ax = 0. The two combine to describe every solution set: if Ax = b is consistent, its solutions are one particular solution plus the entire null space, a translated subspace.

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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 the augmented matrixConsistency, particular solutions and free variables all fall out of the same reduction.
Basis of the column spaceTake the PIVOT columns of the ORIGINAL matrix, not of the reduced one.
Basis of the null spaceParametrise by the free variables; the coefficient vector of each parameter is a basis element.
Write the general solutionx = x_particular + (span of null space basis). Setting all parameters to 0 recovers the particular solution.

Worked example

For A = [[1, 2, 3], [2, 4, 6]], find bases for the column space and the null space.

R2 − 2R1 gives [[1,2,3],[0,0,0]], so there is one pivot, in column 1.
Column space basis: the first ORIGINAL column, (1,2). Rank 1.
Null space: x₁ + 2x₂ + 3x₃ = 0, with x₂ and x₃ free.
Setting (x₂,x₃) = (1,0) gives (−2,1,0); setting (0,1) gives (−3,0,1).

Answer. Column space basis {(1,2)}, null space basis {(−2,1,0), (−3,0,1)} — rank 1 plus nullity 2 equals the 3 columns ✓.

Where marks get dropped

These are the specific errors that cost credit on column space, null space & solution sets questions — QED's rubric penalises each of them separately.

Taking the pivot columns of the REDUCED matrix as a column space basis. Row operations change the column space, so you must return to the original columns.
Forgetting that Ax = b is inconsistent exactly when b is outside the column space.
Describing the solution set of an inhomogeneous system as a subspace. It is a coset — it does not contain 0 unless b = 0.

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Column space, null space & solution sets — frequently asked questions

Why is the solution set a translate of the null space?

If Ax₁ = b and Ax₂ = b then A(x₁ − x₂) = 0, so any two solutions differ by a null space vector. Adding the null space to one solution gives all of them.

How do I test whether b is in the column space?

Row reduce [A | b]. If a row reads 0 … 0 | c with c ≠ 0 then b is outside; otherwise it is in.

What is the row space used for?

It is the orthogonal complement of the null space, which is why the projection onto the row space gives the least-norm solution of an underdetermined system.

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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