Page 240 - A Course in Linear Algebra with Applications

2 2 4 Chapter Seven: Orthogonality in Vector Spaces

Orthogonality and the fundamental subspaces of a
matrix
We saw in Chapter Four that there are three natural sub-
spaces associated with a matrix A, namely the null space, the
row space and the column space. There are of course corre-
T
sponding subspaces associated with the transpose A , so in
all six subspaces may be formed. However there is very little
difference between the row space of A and the column space
T
of A ; indeed, if we transpose the vectors in the row space of
T
A, we get the vectors of the column space of A . Similarly
the vectors in the row space of A T arise by transposing vec-
tors in the column space of A. Thus there are essentially four
interesting subspaces associated with A, namely, the null and
T
column spaces of A and of A . These subspaces are connected
by the orthogonality relations indicated in the next result.

Theorem 7.2.6
Let A be a real matrix. Then the following statements hold:
T ±
(i) null space of A = (column space of A ) ;
1
(ii) null space of A T = (column space of A) -;
7 1
(iii) column space of A = (null space of A ^)- ;
1
(iv) column space of A T = (null space of A) -.
Proof
To establish (i) observe that a column vector X belongs to the
null space of A if and only if it is orthogonal to every column
T ±
T
of A , that is, X is in (column space of A ) . To deduce (ii)
simply replace A by A T in (i). Equations (iii) and (iv) follow
on taking the orthogonal complement of each side of (ii) and
1 1
(i) respectively, if we remember that S = (S- ) - by 7.2.5.