The document provides an overview of vector spaces and related linear algebra concepts. It defines vector spaces, subspaces, basis, dimension, and rank. Key points include: - A vector space is a set that is closed under vector addition and scalar multiplication. It must satisfy certain axioms. - A subspace is a subset of a vector space that is also a vector space. - A basis is a minimal set of linearly independent vectors that span the entire vector space. The dimension of a vector space is the number of vectors in its basis. - The rank of a matrix is the number of linearly independent rows in its row-reduced echelon form. It provides a measure of the matrix's linear

1. Vector Space

2. Hello!

3. History

4. History
1844
Harmann Grassman gave the
introduction of Vector space.

5. History
1888
Guiseppe Peano gave the
definition of vector spaces
and Linear Maps.

6. Vector Space

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.

10. Axioms

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

13. Subspace

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 
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

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
10
00
00
01Sol:

17. Linear Combination

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 

20. 







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
1123
1012
1101
   nEliminatioJordanGuass

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
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0000
1210
1101
321
1
32 vvvw 
t
tctctc  321 ,21,1
(this system has infinitely many solutions)

21. Linear Dependence
&
Independence

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.

26. Basis

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.

29. Example
The vectors e1,e2,…, en are linearly
independent and generate Rn.
Therefore they form a basis for Rn
.

30. Dimension

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."

32. ?
A = [ 1 2 3]
Dimension ?

33. Rank of Matrices

34. Rank
The number of non-zero rows in the
row reduced form of a matrix is
called the row-rank of the matrix.

35. Properties
1. The rank of an m x n matrix is
nonnegative integer and cannot
be greater than either m or n.
That is
rank(A) ≤ min(m,n)

36. Properties
2. If A is a square matrix (i.e. m=n) then
A is invertible if and if only if A has
full rank.

37. Determine the row-rank of
•
row –rank(A)=3

38. ?
A= [1 2 0]
Rank ?

39. Any questions?

40. Thanks!