Real Vector Spaces & Subspaces

Question 1

What is a vector space?

Answer 1

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars in this context.

Question 2

10 Axioms of Vector spaces

Answer 2

1. If u and v are objects in V, then u + v is in V.
2. u + v = v + u
3. u + (v + w) = (u + v) + w
4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V.
5. For each u in V, there is an object −u in V, called a negative of u, such that u + (−u) = (−u) + u = 0.
6. If k is any scalar and u is any object in V, then ku is in V.
7. k(u + v) = ku + kv
8. (k + m)u = ku + mu
9. k(mu) = (km)(u)
10. 1u = u

Question 3

If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisfied.

Answer 3

(a) If u and v are vectors in W, then u + v is in W.
(b) If k is a scalar and u is a vector in W, then ku is in W.

Question 4

What’s the subspace W of V?

Answer 4

If S = {w1, w2, . . . , wr } is a nonempty set of vectors in a vector space V , then the subspace W of V that consists of all possible linear combinations of the vectors in S is called the subspace of V generated by S, and we say that the vectors w1, w2, . . . , wr span W. We denote this subspace as W = span{w1, w2, . . . , wr } or W = span(S)

Question 5

How to determine if 3 vectors span R^3?

Answer 5

We must determine whether an arbitrary vector b = (b1, b2, b3) in R3 can be expressed as a linear combination

b = k1v1 + k2v2 + k3v3

Then check if the system is consistent or inconsistent (Det = 0)