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 .