vector spaces

hello, we just started learning about vector spaces in my linear algebra class and i'm a little confused. I was wondering if someone could give me a link to a good study guide if there is one or if i could get some help doing some examples. i think my problem is i need more examples because the chapter in my book is lacking those in this chapter. I'm going to start with determining vector spaces.

here's an example of what i'm trying to do:

Let V=$\displaystyle R^2$. Define $\displaystyle \oplus$ by (x,y) $\displaystyle \oplus$ (t,s)=(x+t,y+s) and $\displaystyle \odot$ by c $\displaystyle \odot$ (x,y)=(cx,c). Is V with $\displaystyle \oplus$ and $\displaystyle \odot$ a vector space?

I know i'm supposed to use the 10 rules to figure this out but i'm not sure exactly how i'm supposed to go about doing that. any help would be appreciated.

There's lots of study guides on the web as well as books in the library. Checking out the requirements isn't too hard. For instance, is there a 0 vector? Yes, (0,0) satisfies the + requirement. Test this by (0,0) + (x,y) = (x,y) in your addition scheme. Now, is there an inverse to (x,y)? Try (-x,-y). And keep on going for every requirement. The only strange thing about your multiplication by a scalar is that the y coordinate seems to be mapped to 1 for all y.

thankyou very much

It is not a vector space as $\displaystyle \odot$ is not a scalar multiplication, as $\displaystyle c(x_1,y_1)+c(x_2,y_2)\neq c(x_1+x_2,y_1+y_2)$ for $\displaystyle c\neq0$