Chapter 4.1

Question 1

What are the 10 axioms that has to be satisfied in order for a nonempty set of objects V (where addition and multiplication by scalars are defined) to becalled a vector space?

Answer 1

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 in v called the 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 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) (km)u = k(mu)
10) 1u = u

Question 2

What are the four steps to show that a set with two operations is a vector space?

Answer 2

1) Identify the set V of objects that will become vectors
2) Identify the addition and scalar multiplication operations on V
3) Verify axiom 1 and 6. 1 being “closure under addition” and 6 being “closure under scalar multiplication”.
4) Confirm that axioms 2-5 and 7-10 hold

Question 3

Give 5 examples of vector spaces

Answer 3

1) The zero vector space
2) R^(n)
3) R^(infinity)
4) The vector space of mxn matrices
5) The vector space of real valued functions

Question 4

Give two examples of vectorspaces where the axioms fail to hold

Answer 4

1) The vector space of only positive vectors

2) The vector space of R² but where ku = (k u1, 0)

Question 5

Let V be a vector space and u be a vector in v and k a scalar. Then (4)

Answer 5

1) 0u = 0
2) 0k = 0
3) (-1)u = -u
4) if ku = 0 then k = 0 or u = 0