Vectors, operations on vectors, coordinates.
Obvious properties of the operations.
Proposition. If
,
...,
are
linearly dependent then vectors
,
...,
,
,
...,
is
also linearly dependent for arbitrary vectors
,
...,
.
Theorem. Suppose r>n. Then any r vectors
,
...,
in
are linearly dependent.
Example. Vectors
and
are linearly dependent if and only if
.
Theorem. Any set of n linearly independent
vectors in
is a basis of
.
Theorem. Suppose
,
...,
are
linearly independent vectors in
and r<n. Then there exist
vectors
,
...,
such that the set
{
,
...,
,
,
...,
}
is a basis of
.