The document defines key concepts in vector spaces including vector space, subspace, span of a set of vectors, and basis. It provides examples to illustrate these concepts. Specifically: - A vector space is a set of objects called vectors that can be added together and multiplied by scalars, satisfying certain properties. - A subspace is a subset of a vector space that is itself a vector space under the operations of the original space. - The span of a set of vectors S is the set of all possible linear combinations of the vectors in S. - A basis is a set of vectors that spans a vector space and is linearly independent. It provides a standard representation for vectors in the space.

1. • Group Members
• Ravi Gelani (150120116020)
• Simran Ghai (150120116021)

2. Vector space is a system consisting of a set of generalized
vectors and a field of scalars,having the same rules for vector
addition and scalar multiplication as physical vectors and
scalars.
What is Vector Space?
Let V be a non empty set of objects on which the operations of addition
and multiplication by scalars are defined. If the following axioms are
satisfied by all objects u,v,w in V and all scalars k1,k2 then V is called
a vector space and the objects in V are called vectors.

3. 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) There is an object 0 in V, called a zero vector for V, such that
0 + u = u + 0 = u for all u in V
5) For each u in V, there is an object –u in V, called a negative of
u, such that 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
Addition conditions:-

4. Definition:
),,( V : a vector space
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VW
W  : a non empty subset
),,( W ：a vector space (under the operations of addition and
scalar multiplication defined in V)
 W is a subspace of V
Subspaces
If W is a set of one or more vectors in a vector space V, then W is a sub
space of V if and only if the following condition hold;
a)If u,v are vectors in a W then u+v is in a W.
b)If k is any scalar and u is any vector In a W then ku is in W.

5. Every vector space V has at least two subspaces
(1)Zero vector space {0} is a subspace of V.
(2) V is a subspace of V.
 Ex: Subspace of R2
   00,(1) 00
originhethrough tLines(2)
2
(3) R
• Ex: Subspace of R3
originhethrough tPlanes(3)
3
(4) R
   00,0,(1) 00
originhethrough tLines(2)
If w1,w2,. . .. wr subspaces of vector space V then the intersection is
this subspaces is also subspace of V.

6. Example: Set Is Not A Vector

7. Span of set of vectors
If S={v1, v2,…, vk} is a set of vectors in a vector space V,
then the span of S is the set of all linear combinations of
the vectors in S.
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Rcccc ikk  vvv 
If every vector in a given vector space can be written as a linear
combination of vectors in a given set S, then S is called a spanning
set of the vector space.
Definition:

8.  0)((1) span
)((2) SspanS 
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 Notes:
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(a)span (S) is a subspace of V.
(b)span (S) is the smallest subspace of V that contains S.
(Every other subspace of V that contains S must contain span (S).
If S={v1, v2,…, vk} is a set of vectors in a vector space V,
then

9. Basis
• Definition:
 S is called a basis for V
(1) Ø is a basis for {0}
(2) the standard basis for R3:
{i, j, k} i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)
 Notes:
• S spans V (i.e., span(S) = V )
• S is linearly independent
The set of vectors S ={v1, v2, …, vn}V in vector space V is called a
basis for V if ..

10. (3) the standard basis for R
n
:
{e1, e2, …, en} e1=(1,0,…,0), e2=(0,1,…,0), en=(0,0,…,1)
Ex: R4 {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}
Ex: matrix space:
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(4) the standard basis for mn matrix space:
{ Eij | 1im , 1jn }
(5) the standard basis for Pn(x):
{1, x, x2, …, xn}
Ex: P3(x) {1, x, x2, x3}

12. Thank you………..