Page 134 - A Course in Linear Algebra with Applications

118 Chapter Five: Basis and Dimension

namely a cardinal number, which is a sort of infinite analog
of a positive integer. However this goes well beyond our brief,
so we shall say no more about it.
Example 5.1.2
The dimension of R n is n; indeed it has already been shown
in Example 5.1.1 that the columns of the identity matrix I n
n
form a basis of R .
Example 5.1.3
The dimension of P n (R) is n. In this case the polynomials
2
n 1
l,x,x ,... ,x ~ form a basis (called the standard basis) of
P n (R).
Example 5.1.4
Find a basis for the null space of the matrix

A =

Recall that the null space of A is the subspace of R 4
consisting of all solutions X of the linear system AX = 0. To
solve this system, put A in reduced row echelon form using
row operations:

1 0 4/3 4/3'
0 1 1/3 - 2 / 3
0 0 0 0

From this we read off the general solution in the usual way:

( -Ac/3 - 4d/3 •

X =