Properties IV-VI
Exercise 1. Let be a linearly independent system of
vectors in
. Then it can be completed with a vector
to form a basis in
Proof. One way to obtain is this. Let
be a projector onto the span of
and let
Take as
any nonzero vector from the image of
It is orthogonal to any element of the image of
and, in particular, to elements of
Therefore
completed with
gives a linearly independent system
.
is a basis because
for any
Property IV. Additivity. Suppose the th row of
is a sum of two vectors:
Denote
,
(except for the th row, all others are the same for all three matrices). Then
Proof. Denote the system of
vectors
Case 1. If is linearly dependent, then the system of all rows of
is also linearly dependent. By Property III the determinants of all three matrices
are zero and the statement is true.
Case 2. Let be linearly independent. Then by Exercise 1 it can be completed with a vector
to form a basis in
can be represented as linear combinations of elements of
We are interested only in the coefficients of
in those representations. So let
where
and
are linear combinations of elements of
Hence,
We can use Property II to eliminate
and
from the
th rows of
and
respectively, without changing the determinants of those matrices. Let
denote the matrix obtained by replacing the
th row of
with
Then by Property II and Axiom 1
which proves the statement.
Combining homogeneity and additivity, we get the following important property that some people use as a definition:
Property V. Multilinearity. The determinant of is a multilinear function of its rows, that is, for each
it is linear in row
when the other rows are fixed.
Property VI. Antisymmetry. If the matrix is obtained from
by changing places of two rows, then
Proof. Let
,
(all other rows of these matrices are the same). Consider the next sequence of transformations:
By Property II, each of these transformations preserves Recalling homogeneity, we finish the proof.
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