Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
Linear combination
A vector jWi is a linear combination of the vectors jv 1 i; jv 2 i; ...; jv r i if it can be
expressed in the form
jWi ˆ k 1 jv 1 i‡k 2 jv 2 i‡‡k r jv r i;
where k 1 ; k 2 ; ...; k r are scalars. For example, it is easy to show that the vector
jWi ˆ…9; 2; 7† in E 3 is a linear combination of jv 1 i ˆ…1; 2; 1† and
j i ˆ…6; 4; 2†. To see this, let us write
v 2
or
…9; 2; 7† ˆk 1 …1; 2; 1†‡k 2 …6; 4; 2†
…9; 2; 7† ˆ…k 1 ‡ 6k 2 ; 2k 1 ‡ 4k 2 ; k 1 ‡ 2k 2 †:
Equating corresponding components gives
k 1 ‡ 6k 2 ˆ 9; 2k 1 ‡ 4k 2 ˆ 2; k 1 ‡ 2k 2 ˆ 7:
Solving this system yields k 1 ˆ3 and k 2 ˆ 2 so that
jWi ˆ3jv 1 i‡ 2jv 2 i:
Linear independence, bases, and dimensionality
Consider a set of vectors j1i; j2i; ...; ji; r ... jiin n a linear vector space V. If every
vector in V is expressible as a linear combination of j1i; ji; 2 ...; ji; r ...; jni, then
we say that these vectors span the vector space V, and they are called the base
vectors or basis of the vector space V. For example, the three unit vectors
e 1 ˆ…1; 0; 0†; e 2 ˆ…0; 1; 0†, and e 3 ˆ…0; 0; 1† span E 3 because every vector in E 3
is expressible as a linear combination of e 1 , e 2 , and e 3 . But the following three
vectors in E 3 do not span E 3 : j1i ˆ…1; 1; 2†; j2i ˆ…1; 0; 1†, and j3i ˆ…2; 1; 3†.
Base vectors are very useful in a variety of problems since it is often possible to
study a vector space by ®rst studying the vectors in a base set, then extending the
results to the rest of the vector space. Therefore it is desirable to keep the spanning
set as small as possible. Finding the spanning sets for a vector space depends upon
the notion of linear independence.
We say that a ®nite set of n vectors ji; 1 j2i; ...; ji; r ...; jni, none of which is a
null vector, is linearly independent if no set of non-zero numbers a k exists such
that
X n
kˆ1
a k jki
ˆj0i: …5:8†
In other words, the set of vectors is linearly independent if it is impossible to
construct the null vector from a linear combination of the vectors except when all
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