Direct Sum

Definition (Sum of Subspaces).

Let we a vector space
let be subspaces of
The sum of theses subspaces denoted by is the space of all

Note: is a subspace of

Definition (Independence of Subspaces).

The subspace is said to be independent if whenever
with only if

Proposition.

The sum of is the smallest subspace containing all i.e,
if is subspace of and then

Definition (Direct sum).

is said to be the direct sum of if
and are independent
we also represent direct sum by

Proposition.

If is the direct sum of then
can be written uniquely as the sum of

Proposition.

let be subspace of a finite dimensional vector space , and let be the basis for
The order basis obtained by listing the in order is a basis of iff

Corollary.

Let be a subspace of . Then of such that

Proof.
let be a basis of extend this basis to say
let which is linearly independent so it is the basis ow
clearly, and and are independent
□

Definition (Compliment).

' is called a complement of

Note: Compliment is not unique

Proposition.

Let be sub-spaces of finite dimensional vector space then,

Proof.
let be the dimension of subspace and let be the basis of it.
let basis of be
let basis of be
Claim: is the basis of
Clearly spans
lets show that is independent

also
As such that

which implies as is the basis of
so equation ^b488c5 (2.1) becomes

and as is basis of
□
#24-sep

Definition.

Let be a subset of and a compliment of in ,.
For let where
is called the projection of into along

Example

let be the basis of and a basis of

this is true if

Exercise.

Let

let basis of be

we extend it to basis of by adding

Theorem.

Let be a complement of in , ie and define by if for that is is the projection of into along . is a linear transformation.
this is called the projection operator into along
i)
ii)
iii) if
iv) if then
v)

Prove you dumb ass

Exercise.

In the vector space find two diffrent compliment of where is

first lets find the basis of W

we assign arbitrary variable for non pivotal columns

Basis of is

we can see that and are to different basis of compliment