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………..