Colloquium publications.

ANALYSIS SITUS.

19

because C1k-1 contains a 1-cell, ajkl, which does not appear in
any of the circuits CiA, C1, *.. * *, C1k and therefore cannot be
linearly dependent on them. Hence Ci1, C2, * * *, C?' constitute
a complete set of 1-circuits. This sharpens the theorem of ~ 25
a little in that it establishes that there is a complete set of solutions of (Hi) each of which represents a single 1-circuit.
Geometric Interpretation of Matrix Products.
28. According to the definition of multiplication of matrices,
I |aij l I.l - bjk = I ik | I
if and only if
aijbjk = Cik,
j=1
j3 being the number of columns in I laijl | and the number of
rows in I| b I 1.
Hence the equations (Ho) of ~ 19 are equivalent to the matrix
equation,
xi 0
X2 0
IXao 0
in which the matrix on the right has one column containing Ro
zeros.
Since each column of the matrix H, is the symbol (as defined
in ~ 14) for a bounding 0-circuit, (i.e., the jth column is the symbol
for the 0-circuit which bounds aj') any column of Hi is a solution
(xi, x2,.I., Xao) of the set of equations (Ho). By the remark
above we may express this result in the form,
Ho-H1i= 0,
where 0 is the symbol for a matrix all of whose elements are zero.
29. By the boundary of a one-dimensional complex is meant the
set of O-cells each of which is incident with an odd number of
1-cells of the complex. So, for example, a 1-circuit is a linear
graph which has no boundary.