LINEAR ALGEBRA (Math 211)

**Sheet #5**

**Vectors, operations on vectors, coordinates.**

If and then and .

**Obvious properties of the operations.**

In particular , where ; ; ; ...; .

If the vectors are not linearly dependent, they are said to be

**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 .*