The three spaces associated to an mxn matrix A will play a central role: null space, range (also known as column space), and row space.

You will be given a matrix and asked to find bases for all 3 of these spaces.

The dimensions of the spaces are linked in very important ways. It will be very important that your answers do not contradict these fundamental relations.

• Do not confuse a space with its dimension (range vs. rank, null space vs. nullity).
• Remember a basis is a finite set — don’t include free variables in the basis.
• The basis of the zero subspace {0 is the empty set {}, the set with no vectors. This is not the same as the set with the zero vector.

Suppose you are given two vectors v and w and S is the span of {v, w}. Then if you are asked to find an algebraic specification or a minimal set of defining equations, this means that you are looking for equations whose solution set is S.

Example. Let v = [1 1 1]^T , w = [3 —1 2]^T (note these are column vectors). Then the equations with coefficients a₁, a₂, a₃ are the solutions of

a₁ + a₂ + a₃ = 0

3 a₁ - a₂ + 2 a₃ = 0

You can use dimension to tell when you have a basis. For example, if S is the null space of A = [1 2 1]], do v = [1 0 —1 and w = [1 2 —3] form a basis.

• You can tell that S has dimension 2 because the rank of the matrix [1 2 1] is 1. (See the fundamental dimension relations above.)
• It is also easy to check that v and w satisfy Av = 0, Aw = 0, so v and w are in S.
• Finally, it is not hard to check that v and w are independent, so the set {v, w} is an independent set of two vectors in a two-dimensional space S. So the set of vectors spans a two-dimensional subspace of S, which is S itself.