1. ## Linear Subspace problem.

I am very sorry if I am bothering you guys a lot but I am trying to understand some of this math and its very hard to do it alone

So I have this problem that says: Find which of the following sets are linear subspaces of the corresponding linear spaces:

I am only going to show the first one because maybe if I understand it I can do the rest(I have the answers but I have no idea how it comes out). This is the problem:

$\displaystyle X_{0}=\left \{ (x1,...\: xn)/ x1+...+xn=0 \right.\left. \right \} \subset \mathbb{R}^{n}, (\mathbb{R}^{n}, \mathbb{R}, \cdot )$

Now I understand to be a linear subspace the set must contain the null vector, be closed under multiplication and addition. My problems begin from the step that I have no idea what $\displaystyle (\mathbb{R}^{n} , \mathbb{R}, \cdot )$ is.

Also are these the vectors within $\displaystyle (x1,...\: xn)$ the set X and if so what does this do: $\displaystyle / x1+...+xn=0$

2. $\displaystyle \mathbb{R}$ is the set of real numbers.

$\displaystyle \mathbb{R}^n$ is the set of n-tuples of real numbers.

I would assume that the notation $\displaystyle (\mathbb{R}^n, \mathbb{R}, \cdot )$ means that you're looking at the vector space of n-tuples of real numbers over the field of reals where the operation is normal multiplication.

Now, the null vector is in the subset because $\displaystyle 0+\cdots +0 = 0$

If $\displaystyle (x_1,\ldots ,x_n), (y_1,\ldots ,y_n)$ are in the subset, then
$\displaystyle x_1+y_1+\ldots x_n+y_n = x_1+\ldots +x_n+ y_1+\ldots +y_n=0+0=0$.
So $\displaystyle (x_1,\ldots ,x_n) + (y_1,\ldots ,y_n)$ is in the subset.

Scalar multiplication is similar.

Yes, it's a subspace.

3. Ok I think am getting it but whats confusing is that my book talked only about "internal binary operation on V(I guess in this case would be $\displaystyle {R}^n$), called addition and denoted by +, such that (V,+) is a commutative group". So you can only imagine when they decided to throw in a * how confused I got.

But am I correct to assume that is the same thing but instead of + its *?

4. To be honest, that notation is quite confusing. I'm not sure why they chose those 3 things to emphasize (as opposed to addition, for example). I wouldn't worry too much about it though - I think the question is pretty clear. Maybe ask your teacher why this notation is being used.

5. It is somewhat surprising to lean that mathematical notation is anything but standard. There is a series of probability books all by the same author in which notation varies from book to book. That is why is almost pointless to ask about notation is a forum such as this.
Sometimes if you give the title and author it is possible that someone may know that particular textbook.

That said, I think Steve is correct in his guess.
$\displaystyle \left( {\mathbb{R}^n ,\mathbb{R}, \cdot } \right)$ tells us that the v-space is the set of real n-tuples with vector addition, the scalar field is $\displaystyle \mathbb{R}$ and $\displaystyle \cdot$ tells us the scalar multiplication is that of the real numbers.

But that is surely explained in the text material.