Multilinearity in columns
The axioms and properties of determinants established so far are asymmetric in that they are stated in terms of rows, while similar statements hold for columns. Among the most important properties is the fact that the determinant of is a multilinear function of its rows. Here we intend to establish multilinearity in columns.
Exercise 1. The determinant of is a multilinear function of its columns.
Proof. To prove linearity in each column, it suffices to prove additivity and homogeneity. Let be three matrices that have the same columns, except for the
th column. The relationship between the
th columns is specified by
or in terms of elements
for all
Alternatively, in Leibniz' formula
(1)
for all such that
we have
(2)
If for a given set
there exists only one
such that
Therefore by (2) we can continue (1) as
Homogeneity is proved similarly, using equations or in terms of elements
for all
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