Page 146 - A Course in Linear Algebra with Applications

130 Chapter Five: Basis and Dimension

Now by 5.2.1 and 5.2.3 the row spaces of A and N have the
same dimension, with a like statement for column spaces. But
it is clear from the form of N that the dimensions of its row
and column spaces are both equal to r, so the result follows.
It is a consequence of 5.2.4 that every matrix has a unique
normal form; for the normal form is completely determined
by the number of l's on the diagonal.

The rank of a matrix
The rank of a matrix is defined to be the dimension of the
row or column space. With this definition we can reformulate
the condition for a linear system to be consistent.

Theorem 5.2.5
A linear system is consistent if and only if the ranks of the
coefficient matrix and the augmented matrix are equal.

This is an immediate consequence of 2.2.1, and 5.2.2.

Finding a basis for a subspace
Suppose that X\, X2, • • •, -Xfc are vectors in F n where F
is a field. In effect we already know how to find a basis for
the subspace generated by these vectors; for this subspace is
simply the column space of the matrix [Xi|X 2 | .. \X^]. But
.
n
what about subspaces of vector spaces other than F ? It turns
out that use of coordinate vectors allows us to reduce the
n
problem to the case of F .
Let V be a vector space over F with a given ordered
basis vi, v 2 ,..., v n , and suppose that S is the subspace of
V generated by some given set of vectors w 1 ) W2,...,w m .
The problem is to find a basis of S. Recall that each vec-
tor in V has a unique expression as a linear combination of
the basis vectors v i , . . . , v n and hence has a unique coordi-
nate column vector, as described in 5.1. Let w, have co-
ordinate column vector Xi with respect to the given basis.
Then the coordinate column vector of the linear combination