LINEAR ALGEBRA (Math 211)
Sheet #5
Vectors, operations on vectors, coordinates.
An n-dimensional vector space is the set of all ordered
n-tuples of real numbers:
If
and
then
and
.
Obvious properties of the operations.
Linear combination
.
In particular
,
where
;
;
;
...;
.
Linear independence.
Definition. Vectors
,
,
...,
are
said to be linearly dependent if the zero vector
is a nontrivial linear combination of
,
...,
.
This means, there exist r numbers
,...,
not all zero such that
.
If the vectors are not linearly dependent, they are said to be
linearly independent.
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
.
Basis.
Definition. A set of vectors
,
...,
is
called a basis of the space
if for every
there is a unique linear combination of
,
...,
which is equal to .
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 .