Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR INDEPENDENCE, BASES, AND DIMENSIONALITY
the coecients vanish. For example, non-zero vectors j1i and j2i of E 2 that lie
along the same coordinate axis, say x 1 , are not linearly independent, since we can
write one as a multiple of the other: j1i ˆ aj2i, where a is a scalar which may be
positive or negative. That is, j1i and j2i depend on each other and so they are not
linearly independent. Now let us move the term ajito 2 the left hand side and the
result is the null vector: j1i ajiˆ 2 j0i. Thus, for these two vectors j1i and j2i in
E 2 , we can ®nd two non-zero numbers (1, a† such that Eq. (5.8) is satis®ed, and
so they are not linearly independent.
On the other hand, the n vectors j1i; ji; 2 ...; ji; r ...; jniare linearly dependent if
it is possible to ®nd scalars a 1 ; a 2 ; ...; a n , at least two of which are non-zero, such
that Eq. (5.8) is satis®ed. Let us say a 9 6ˆ 0. Then we could express j9i in terms of
the other vectors
jiˆ 9
Xn
iˆ1;6ˆ9
That is, the n vectors in the set are linearly dependent if any one of them can be
expressed as a linear combination of the remaining n 1 vectors.
a i
a 9
ji: i
Example 5.4
The set of three vectors j1i ˆ…2; 1; 0; 3†, jiˆ…1; 2 2; 5; 1†; j3i ˆ…7; 1; 5; 8† is
linearly dependent, since 3ji‡2
1 j ij3i ˆ j0i.
Example 5.5
The set of three unit vectors je 1 i ˆ…1; 0; 0†; je 2 i ˆ…0; 1; 0†, and je 3 i ˆ…0; 0; 1† in
E 3 is linearly independent. To see this, let us start with Eq. (5.8) which now takes
the form
or
from which we obtain
a 1 je 1 i‡ a 2 je 2 i‡ a 3 je 3 i ˆ ji 0
a 1 …1; 0; 0†‡a 2 …0; 1; 0†‡a 3 …0; 0; 1† ˆ…0; 0; 0†
…a 1 ; a 2 ; a 3 †ˆ…0; 0; 0†;
the set of three unit vectors je 1 i; je 2 i, and je 3 i is therefore linearly independent.
Example 5.6
The set S of the following four matrices
j1i ˆ 1 0
; jiˆ 2 0 1
0 0
0 0
; j3i ˆ 0 0
1 0
; j4i ˆ 0 0
;
0 1
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