0.1 Vector spaces

Question 1

What is a vector?

Answer 1

A vector is anything with magnitude and direction.

Question 2

What is a vector space?

Answer 2

A vector space is a mathematical structure that allows us to study and manipulate vectors.

A set of vectors where they can be added to each other or multiplied by scalars.

Question 3

What are the two possible operations in a vector space?

Answer 3

Vector addition and scalar multiplication.

Question 4

What are the axioms of a vector space?

Answer 4

1. u + v = v + u
2. u + (v + w) = (u + v) + w
3. For all u belonging to V there is -u belonging to V such that -u + u = 0
4. There exist 0 in V such that u + 0 = u
5. There exist 1 in the field such that 1.u = u
6. alpha(u + v) = alpha u + alpha v
7. u (alpha + beta) = alpha u + beta u
8. u (alpha beta) = (u alpha) beta.

Question 5

What are some examples of vector spaces?

Answer 5

1. The set R is a vector space over R.
2. The set R^2 of all ordered pairs of real numbers is a vector space over R.