7. Vector Space
A Vector space V is a set that is
closed under finite vector addition
and scalar multiplication.
8. The set of all Integers is not a vector space.
1 ϵ V, ½ ϵ R
(½ ) (1) = ½ !ϵ V
It is not closed under scalar multiplication
9. The set of all second degree
polynomials is not a vector space
Let
P(x)=-X
Q(x)=2X+1
=>P(x)+Q(x)=X+1 ₵ V
It is not closed under scalar multiplication.
11. The axioms need to be satisfied to be a
vector space:
•Commutivity:
X+Y=Y+X
•Associativity:
(X+Y)+Z=X+(Y+Z)
•Existence of negativity:
X+(-X)=0
•Existence of Zero:
X+0=X
12. The axioms need to be satisfied to be a
vector space:
•Associativity of Scalar multiplication:
(ab)u=a(bu)
•Right hand distributive:
k(u+v)=ku+kv
•Left hand distributive:
(a+b)u=au+bu
•Law of Identity:
1.u=u
14. Subspace
If W is a nonempty subset of a vector space V,
then W is a subspace of V
if and only if the following conditions hold.
15. Conditions
(1) If u and v are in W, then u+v is in W.
(2) If u is in W and c is any scalar, then cu is in W.
16. WBA
10
01
222 ofsubspaceanotis MW
Ex: The set of singular matrices is not a subspace of M2×2
Let W be the set of singular matrices of order 2. Show that
W is not a subspace of M2×2 with the standard operations.
WB,WA
10
00
00
01Sol:
18. Linear Combination
A vector v in a vector space V is called linear combination
of the vectors u1, u2, u3, uk in V if v can be written in the
form
v=c1u1+c2u2+…+ckuk
where c1c2,…,ck are scalars
19. 123
12
1
321
21
31
ccc
cc
cc
Ex : Finding a linear combination
,,ofncombinatiolinearais(1,1,1)Prove
1,0,1)((0,1,2)(1,2,3)
321
321
vvvw
vvv
Sol: 332211(a) vvvw ccc
1,0,12,1,03,2,11,1,1 321 ccc
)23,2,( 3212131 ccccccc
22. Linear Dependence
Let a set of vectors S in a vector space V
S={v1,v2,…,vk}
c1v1+c2v2+…+ckvk=0
If the equations has only the trivial solution
(i.e. not all zeros) then S is called linearly dependent
23. Example
Let
a = [1 2 3 ] b = [ 4 5 6 ] c= [5 7 9]
Vector c is a linear combination of
vectors a and b, because c = a + b.
Therefore, vectors a, b, and c is linearly
dependent.
24. Linear Independence
Let a set of vectors S in a vector space V
S={v1,v2,…,vk}
c1v1+c2v2+…+ckvk=0
If the equations has only the trivial solution
(c1 = c2 =…= ck =0) then S is called linearly independent
25. Example
Let
a = [1 2 3 ] b = [ 4 5 6 ]
Vectors a and b are linearly
independent, because neither vector is
a scalar multiple of the other.
27. Basis
A set of vectors in a vector space V is called a basis if the
vectors are linearly independent and every vector in the
vector space is a linear combination of this set.
28. Condition
Let B denotes a subset of a vector space V.
Then, B is a basis if and only if
1. B is a minimal generating set of V
2. B is a maximal set of linearly independent
vectors.
31. Dimension
The number of rows and columns of a matrix,
written in the form rows×columns.
If matrix below has m rows and n columns, so
its dimensions are m×n. This is read aloud,
“m by n."
