Linear Algebra, Notes 2
Def 4.4 A real
vector space
is a set V of elements on which we have two operations +
and * defined with these properties:
(a) if u, v are elements in V , then u+v is in V (closed under +):
(i) u+v = v+u for all u, v in V
(ii) u+(v+w) = (u+v)+w for u, v, w in V
(iii) there exists an element 0 in V such that u+0 = 0+u = 0 for u in V.
(iv) for each u in V there exists an element u in V such that u+(u) = (u)
+u = 0.
(b) If u is any elemnt in V and c is a real number, then c*u (or cu) is in V (V is
closed under scalar multiplication).
(i) c*(u+v) = c*u + c*v for any u,v in V, c a real number
(ii) (c+d) * u = c*u + d*u for any u in V, c, d real numbers
(iii) c * (d*u) = (cd) * u for any u in V, c, d real numbers
(iv) 1*u = u for any u in V
Theorem 4.2
If V is a vector space, then
(a) 0*u = u for any u in V
(b) c*0 = 0 for any scalar c
(c) if c*u = 0, then either c = 0 or u = 0.
(d) (1)*u = u for any vector u in V
Def 4.5 Let V be a vector space and W a nonempty subset of V. If W is a vector space
with respect to the operations in V, then W is called a
subspace
of V.
Theorem 4.3
Let V be a vector space and let W be a nonempty subset of V. Then W
is a subspace of V if and only if the following conditions hold:
(a) if u, v are in W, then u+v is in W
(b) if c is a real number and u is any vector in W, then c*u is in W.
Definition 4.6 Let v
1
, v
2
, ..., v
n
be vectors in vector space V. A vector V is a
linear
combination
of v
1
, v
2
, ..., v
n
, if
v = a
1
v
1
+ a
2
v
2
+ ... + a
n
v
n
Definition 4.7 If S = {v
1
, v
2
, ..., v
n
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 Spring '08
 comech
 Linear Algebra, Algebra, Vector Space, Rn Theorem, A. Theorem, W. Theorem

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