MATRICES AND VECTOR SPACES
In a similar way we may denote a vectorxin terms of its componentsxiin abasisei,i=1, 2 ,...,N, by the array
x=
x 1
x 2
..
.
xN
,which is a special case of (8.25) and is called acolumn matrix(or conventionally,
and slightly confusingly, acolumn vectoror even just avector– strictly speaking
the term ‘vector’ refers to the geometrical entityx). The column matrixxcan also
be written as
x=(x 1 x 2 ··· xN)T,which is thetransposeof arow matrix(see section 8.6).
We note that in a different basise′ithe vectorxwould be represented by adifferentcolumn matrix containing the componentsx′iin the new basis, i.e.
x′=
x′ 1
x′ 2
..
.
x′N
.Thus, we usexandx′to denote different column matrices which, in different bases
eiande′i, represent thesamevectorx. In many texts, however, this distinction is
not made andx(rather thanx) is equated to the corresponding column matrix; if
we regardxas the geometrical entity, however, this can be misleading and so we
explicitly make the distinction. A similar argument follows for linear operators;
the same linear operatorAis described in different bases by different matricesA
andA′, containing different matrix elements.
8.4 Basic matrix algebraThe basic algebra of matrices may be deduced from the properties of the linear
operators that they represent. In a given basis the action of two linear operators
AandBon an arbitrary vectorx(see the beginning of subsection 8.2.1), when
written in terms of components using (8.24), is given by
∑
j(A+B)ijxj=∑jAijxj+∑jBijxj,∑j(λA)ijxj=λ∑jAijxj,∑j(AB)ijxj=∑kAik(Bx)k=∑j∑kAikBkjxj.