Prove or disprove that a given subset W of is a subspaceSol

. Prove or disprove that a given subset W of is a subspace

Solution

Let us begin with the definition of a subspace of a vector space.

Definition. Let be a vector space over a field and be a subset of . Then, is said to be a subspace of if and only if it satisfies the following properties:

(NOTE: In order to prove a subset to be a subspace, we need to prove that it satisfies the above three properties. However, to prove that a subset is not a subspace, we need to present a counter- example that violates any one( or more) of the above three properties.)

Proof.

Here, is a subset of . Therefore, every element is a 3-tupple. This means that the zero vector of will be . For any element to belong , it must satisfy the following inequality:

     (According to the definition of )

                                    Clearly, Therfore, .

                            

                                    (Note: means ‘less than OR equal to’. For example, , .)

                            

(To show: )

                                  … (1)

                                           
                               … (2)

Consider

Adding (1) and (2) we get, .

This shows that .

(To show: )

                                  … (3)

From (3) we get,

This shows that .

             

              As, satisfies the three properties, it is a subspace of .

Proof.

Here, is a subset of . Therefore, every element is a 3-tupple. This means that the zero vector of will be . For any element to belong , it must satisfy the following inequality:

    (According to the definition of )

                                    Clearly, Therfore, .

                                                         

(To show: )

                                … (1)

                                           
                              … (2)

Consider

Now consider

                                                                  (On rearranging)

                                                                                                      (from (1) and (2))

                                                                                                  

This shows that .

(To show: )

                                … (3)

From (3) we get,

This shows that .

             

              As, satisfies the three properties, it is a subspace of .