8. Row Space, Column Space, Nullspace

8 Row space, column space, nullspace (kernel)

8.1 Subspaces The 2-dimensional plane, 3-dimensional space, Rn, Rm×n (m × n matrices) are all examples of Vector spaces. They all have the linear combination property: in any of these spaces, for any list X1,...,Xk of points from that space, and real numbers α1,...,αk, the space contains the linear combination α1X1 + ... + αkXk Other examples of vector subspace include

• Any plane through O in 3 dimensions and similar...

• We call the elements of a vector space points or vectors.

• For any list X1,...,Xk of points, the set of all linear combinations

α1X1 + ... + αkXk

is a vector space in its own right. This set of all linear combinations is called the subspace spanned by the points X1,...,Xk.

• For any m × n matrix A, there are three notable spaces associated with A:

• the set of all linear combinations of rows of A, called the row space of A. • the set of all linear combinations of columns of A, called the column space of A. • The set of all column vectors X (in Rn×1) such that AX = O. This is called the kernel or nullspace of a.

Row space, column space, and kernel (nullspace) of a matrix are all examples of ‘subspaces.’ The row space of Am×n is the space spanned by its rows and the column space is the space spanned by its columns. ‘Vectors’ usually mean column vectors. If we need to distinguish between row and column vectors, we write R1×n for the set of row vectors of width n, and Rm×1 for the column vectors of height m. It follows that the column space of A is

{AX : X ∈ Rn×1}

and {XA : X ∈ R1×m} is the row space. The kernel or nullspace is the set of solutions (in Rn×1) to

AX = O.

1 8.2 Bases for row, column, and null spaces (kernel) It can be shown that every vector space has a basis (or equivalently, every vector space can be given a coordinate system). The object is to construct bases for these three spaces. This is an exercise in ...well, constructing bases. The main interest in constructing bases will be to construct new coordinate systems which allow certain problems to be simpliﬁed. The important facts are

• All bases for a given subspace have the same cardinality. For instance, every basis for R3 must contain 3 (column) vectors.

• The number of vectors in a basis for a subspace — the same for every basis — is called the dimension of the subspace.

• Given a matrix Am×n, its row space and column space have the same dimension.

• Given that the row space and column space have dimension r, the kernel has dimension n − r.

8.3 Row space, column space, and kernel (nullspace) Given an m × n matrix A, • The column space of A is the subspace spanned by its columns. Or, equivalently, {AX : X ∈ Rn}

• The row space is the subspace spanned by its rows.

• The kernel or nullspace is {X ∈ Rn : AX = O}.

We can get bases for the row and column spaces from the RREF, and also for the kernel.

• Let F be the RREF of A.

• The nonzero rows of F are a basis for the row space.

• Columns of A corresponding to the leading columns of F are a basis for the column space of A.

• A basis for the kernel can be constructed from the RREF.

Example -1 -3 -2 -2 -4-19*-1=R1 -3 -9 -6 -8-14-67 +3R1 1 3 2 3 627 -R1 000128

2 -1 -3 -2 -4 -6-29 +R1

1 3 2 2 419 -2*R2 0 0 0 -2 -2-10 /(-2)=R2 0 0 0 1 2 8-R2 0 0 0 1 2 8-R2 0 0 0 -2 -2-10 +2*R2

132029-2*R3 132003 rref 000115-R3 000102 000013=R3 000013 000013-R3 000000 000000 000000

Row-space basis [1, 3, 2, 0, 2, 9], [0, 0, 0, 1, 1, 5], [0, 0, 0, 0, 1, 3]; nonzero rows of RREF F of original matrix A.

Leading columns 1, 4, 5. Column-space basis: columns 1, 4, 5 of A.

1 4 5  −1   −2   −4   −3   −8   −14    ,   ,    1   3   6         0   1   2         −1   −4   −6  To get a basis for the nullspace:

• The nullspace of A is that of F .

• Each nonzero row of F can be interpreted as expressing a ‘leading variable’ in terms of the ‘nonleading variables.’

• We think of the nonleading variables in a vector in the kernel as ‘independent variables.’ We don’t have to, but it seems neater if we relabel these variables as r,s,t,u,... instead

of xi1 ,xi2 ...

• Then we can express the ‘leading variables’ in terms of the others.

3 Taking the nonzero rows of F , the nonleading variables are x2, x3, and x6.

132003  000102   000013  x2 = r, x3 = s,x6 = t x1 +3r +2s +3t =0: x1 = −3r − 2s − 3t ...x4 = −2t x5 = −3t

So the kernel can be described as −3 −2 −3  1   0   0   0   1   0  r   + s   + t    0   0   −2         0   0   −3         0   0   1  and the three column-vectors shown are linearly independent and form a basis for the kernel (or nullspace). We started with a 5 × 6 matrix: m = 5,n = 6. Bases for row and column space both contained r = 3 vectors, and the kernel k = 3 also, consistent with the formula n − r = k.

(8.1) Proposition As predicted earlier, the row and column spaces have the same dimension, call it r, and the kernel has ‘complementary dimension’ n − r.