Vector Spaces,subspaces,Span,Basis
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Transcript of Vector Spaces,subspaces,Span,Basis
VCLA (2110015)
Active Learning Assigment
Branch-ITDiv:-D_DG1
• Group Members• Ravi Gelani (150120116020)• Simran Ghai (150120116021)
TOPIC:-VECTOR SPACES , SUBSPACES , SPAN , BASIS
GUIDED BY:- PROF.SIKHA YADAV
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.
1) If u and v are objects in V, then u + v is in V2) u + v = v + u3) u + (v + w) = (u + v) + w4) There is an object 0 in V, called a zero vector for V, such that
0 + u = u + 0 = u for all u in V5) For each u in V, there is an object –u in V, called a negative of
u, such that u + (-u) = (-u) + u = 06) If k is any scalar and u is any object in V then ku is in V7) k(u+v) = ku + kv8) (k+l)(u) = ku + lu9) k(lu) = (kl)u10) 1u = u
Addition conditions:-
Definition:),,( V : a vector space
VWW : 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.
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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
0 0, (1) 00origin he through tLines (2)
2 (3) R
• Ex: Subspace of R3
origin he through tPlanes (3)3 (4) R
0 0, 0, (1) 00
origin he through tLines (2)
If w1,w2,. . .. wr subspaces of vector space V then the intersection is this subspaces is also subspace of V.
Example: Set Is Not A Vector
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.
)(Sspan )in vectorsof nscombinatiolinear all ofset (the
2211
SRcccc 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:
0)( (1) span
)( (2) SspanS
)()( , (3)
2121
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SspanSspanSSVSS
Notes:
VSSV
V SVS
ofset spanning a is by )(generated spanned is
)(generates spans )(span
(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
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 ..
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(3) the standard basis for Rn :
{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:
1000
,0100
,0010
,0001
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(4) the standard basis for mn matrix space:
{ Eij | 1im , 1jn }
(5) the standard basis for Pn(x):
{1, x, x2, …, xn}Ex: P3(x) {1, x, x2, x3}
Thank you………..