dimension, basis watch

1. I have some questions ......

I know dimension= number n of vectors in a basis of a finite dimension vector space V, s.t. dim(V)=n

So, is dim(R^n)=n??

Also,

Let S be a sequence of vectors in R^3.

1)If S=(1,0,0),(1,1,1),(0,0,0),
then S is not a basis of R^3 because of (0,0,0)...... Am I right? Is that true in all cases if there are 3 vectors and one of them is (0,0,0)?

2)If there are only 2 vectors in S, then they can't form a basis of R^3.....Right?
What if there are 4 vectors that are linear independent?

It would be helpful if you can give me so simple examples=)

Thanks a lot!!!
2. 1) Yes you're right. Yes it's true in all the cases if one vector is (0,0,0) since that set is not linearly independent and thus cannot be a basis.
2) Yes right. If there are 4 vectors linearly independent you know that there's a smaller set of vectors which would give you the same information about the space you are considering and its dimension.
3. (Original post by gbongibo20)
1) Yes you're right. Yes it's true in all the cases if one vector is (0,0,0) since that set is not linearly independent and thus cannot be a basis.
2) Yes right. If there are 4 vectors linearly independent you know that there's a smaller set of vectors which would give you the same information about the space you are considering and its dimension.
Thanks!
But I still got a question,,,,
in R^3, do the vectors need to be in the form (a,b,c)?
in R^4, do the vectors need to be in the form (a,b,c,d)?....etc
Do you understand?
4. I don't know if I get what you mean but yes depending on the dimension of the field/space you are considering its elements will have as much elements as the dimensions of the space
5. Read up on positive definite matrices and inner product spaces. Also spanning sets. If you want to know more.
6. If you think you'll follow the hint by marinade I would suggest "Linear Algebra", Gilbert Strang
7. For 2) if you’re talking about S and R^3, then you can’t have 4 linearly independent vectors.
8. Or if you don't want a book on linear algebra (or other higher topics) you could just read this

https://en.wikipedia.org/wiki/Dimens..._vector_spaces

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