https://steemit.com/mathematics/@drifter1/mathematics-linear-algebra-vector-spaces
real vector spaces
(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) 0 (Zero vector) : 0+u = u+0 = u for all u in V
(5) -u (a negative of u) : 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+l)u = ku+lu
(9) k(lu) = (kl)u
(10) 1u = u
It is possible for one vector space to be contained within another.
Definition
A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication on V.
general, to show that a nonempty set W with two operations is a vector space one must verify the ten vector space axioms.
but if W is a subspace of a known vector space V, then certain axioms need not be verified because they are "inherited" from V.
theorem 4.2.1
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 hold.
prove
(a) If u and v are vectors in W, then u+v is in W.
(b) If k is any scalar and u is any vector in W, then ku is in W.
EXAMPLE1
Let W be any plane through the orgin and let u and v be any vectors in W.
Is W a subspace of R^3?
#### 모든 K1, K2에 대해서 만족해야한다!!!
theorem 4.2.2
if W1,W2,W3,.....,Wr are subspaces of a vector space V, then the inter-section of these subspaces is alo a subspace of V.
