quiz 5 Flashcards
how many vector space axioms are there?
10
what is axiom 1?
the sum of u and v, denoted by u + v, is in V
what is axiom 2?
u + v = v + u
what is axiom 3?
(u + v) + w = u + (v + w)
what is axiom 4?
there is a zero vector 0 in V such that u + 0 = u
what is axiom 5?
for each u in V, there is a vector -u in V such that u + (-u) = 0
what is axiom 6?
the scalar multiple of u by c, denoted by cu, is in V
what is axiom 7?
c(u + v) = cu + cv
what is axiom 8?
(c + d)u = cu + du
what is axiom 9?
c(du) = (cd)u
what is axiom 10?
1u = u
what is the definition of a vector space?
the 10 axioms
what are the 10 axioms in order?
1) the sum of u and v, denoted by u + v, is in V
2) u + v = v + u
3) (u + v) + w = u + (v + w)
4) there is a zero vector in V such that u + 0 = u
5) for each u in V, there is a vector -u in V such that u + (-u) = 0
6) the scalar multiple of u by c, denoted by cu, is in V
7) c(u + v) = cu + cv
8) (c + d)u = cu + du
9) c(du) = (cd)u
10) 1u = u
a subspace of a vector space V is a subset H of V that has _____
three properties
what are the properties to be a subspace?
1) the zero vector of V is in H
2) H is closed under vector addition. That is, for each u and v in H, the sum u + v is also in H
3) H is closed under multiplication. That is, for each u in H and each scalar c, the vector cu is in H
