Proving Nonempty Sets are Subspaces & Examples

[zxc210188]

(1)Prove that a nonempty set W is a subspce of a vector space V iff a x+b y is an element of W for all scalars a and b and all vectors x and y in W

(2)Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspaces of V

 

[Hurkyl]

Have you had any thoughts at all on the problem? Really, (1) nothing more than an exercise in definitions, and you almost have to try in order to get (2) wrong by guessing at an answer.

 

[HallsofIvy]

What is the DEFINITION of "subspace"? Show that if "ax+ by is in W for all numbers a and b and vectors x and y in W" is true then each of the conditions for a subspace are satisfied. Because this is an "if and only if" statement, you need to then turn around and show that, if W satisfies all the conditions for a subspace, then "ax+ by is in W for all numbers a and b and vectors x and y in W" is true.

As for the second part, as Hurkyl said, it's almost impossible to get it wrong! To make it as easy as possible use R² as V and choose 2 very easy subspaces.

 

[zxc210188]

I know it should be prove in two direction, but I have no idea about how to prove it.
I am a student in Taiwan, so my English is not very well. I cannot understand what the textbook talk about compeletely, so I learn it from doing questions and problems, I did try to think on the problems, but I failed.It may be easy for you but hard for a foreign student.