# How to prove it

1. Nov 25, 2006

### 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

Last edited: Nov 25, 2006
2. Nov 25, 2006

### Hurkyl

Staff Emeritus
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.

3. Nov 25, 2006

### 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.

4. Nov 25, 2006

### 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 text book 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.

Last edited: Nov 25, 2006