What Are the Key Vector Spaces You Should Understand?

[Zhalfirin88]

This isn't a homework question, but I thought it'd still be an appropriate place.

So this is about vector spaces. I know these are sort of abstract spaces but I'd like more explanation on them.

1) $$\mathbb{R}$$ This is the space of real vectors right, like from real numbers?

2) $$\mathbb{P}$$ What does this mean that it's a polynomial space? Like the standard basis is {1,t,t²,t³...} but how is $$\mathbb{P}_n$$ different from $$\mathbb{R}^n$$

3) $$\mathbb{C}$$ My instructor basically showed us this one, but didn't explain it cause we don't use it in our linear algebra class.

Edit: Are there any other majors spaces that I didn't list but are important to know?

 

[Mark44]

This isn't a homework question, but I thought it'd still be an appropriate place.

So this is about vector spaces. I know these are sort of abstract spaces but I'd like more explanation on them.

1) $$\mathbb{R}$$ This is the space of real vectors right, like from real numbers?

Yes, pretty much. A vector in this space extends from the origin to a given real number. A vector space consists, naturally enough, of vectors, whose components come from some field. In this case, the field is the real numbers. Scalar multiplication means multiplication by real numbers.

2) $$\mathbb{P}$$ What does this mean that it's a polynomial space? Like the standard basis is {1,t,t²,t³...} but how is $$\mathbb{P}_n$$ different from $$\mathbb{R}^n$$

Usually you'll see this as P_n, the (function) space of polynomials of degree <= n. A function space is structurally identical to a vector space, and must satisfy the same 10 axioms. Notice that a basis for Pⁿ consists of n + 1 functions, while a basis for Rⁿ consists of n vectors.

3) $$\mathbb{C}$$ My instructor basically showed us this one, but didn't explain it cause we don't use it in our linear algebra class.

This would be similar in some respects to the vector space R, but the field is the complex numbers. The components are complex numbers, and scalar multiplication is done using complex numbers.

Edit: Are there any other majors spaces that I didn't list but are important to know?

Matrices of a given size form a vector space. E.g., there's a vector space that consists of 2 x 2 matrices, another for 3 x 2 matrices, and so on. For more examples, see http://en.wikipedia.org/wiki/Examples_of_vector_spaces.

 

[Zhalfirin88]

Thanks Mark. :)