Span Vector

Span of Vector Space

Tarkeshar Singh

Department of MathematicsBITS Pilani KK Birla Goa Campus, Goa

12th Feb. 2015

Linear Combination

Definition

A vector w V is called a linear combination of vectorsv1, v2, . . . , vr if it can be written as

w =r

i=1

kivi.

where ki are scalars.

Definition

Let vi = (ai ,1, ai ,2, , ai ,n) where ai ,j are reals for i = 1, 2, ,mand j = 1, 2, , n. Thenx =

mi=1 bi vi = (b1a1,1 + + bmam,1, , b1a1,n + + bmam,n)

is called a linear combination of the vectors v1, v2, , vm wherebks are also real numbers.

Tarkeshar Singh Linear Algebra

Examples

Example

Let u = (1, 2,1) &v = (6, 4, 2) R. Check whetherw1 = (9, 2, 7) is a linear combination of the vectors u and v.

Example

Is x = (4,1, 8) a linear combination of the vectors u and v?


Remark

If the linear system is consistent then a given vector can be writtenas a linear combination of others vectors. If the linear system isnot consistent then the given vector can not be written as a linearcombination of other vectors.

So we have seen that w =r

i=1 kivi for some vector w V andx 6=

ri=1 kivi for some vector x V .


If W is a collection of all such vectors w V then W is asubspace of the vector space V .


Subspaces

Now, it is time to recall the following tests for subspace.

Theorem

A subset W of a vector space V is a subspace iff W is closedunder vector addition and scalar multiplication.

Theorem

Let W be a subspace of a vector space V . Then for a1 an R3and w1, , wn W , we have a1w1 + + anwn W .


Subspace

Theorem

If v1, v2, . . . , vr are vectors in a vector space V then

The set W of all linear combination v1, v2, . . . , vr is asubspace of V .

W is the smallest subspace of V that contains v1, v2, . . . , vr.


Span

Definition

Let S be a nonempty subset of a vector space V . Then the spanof S in V is the set of all possible finite linear combination ofvectors of S and is denoted by span(S).

Now, we will look into the following theorem.

Theorem

Let S be a nonempty subset of a vector space V . Then

1 S span(S).2 Span(S) is a subspace of V .

3 If W is any subspace of V with S W , then span(S) W .4 Span(S) is the smallest subspace of V containing S.


Span

Corollary

Let S1,S2 be two subset of a vector space V with S1 S2. Thenspan(S1) span(S2).


Simplifying span

Remark

The span of an empty set is {0}.


CMYK Color Model

Color magazines and books are printed using this model,where C, M, Y, K are some colors.

Colors can be created either by mixing inks of the four typesand printing with these mixed inks, which is known as TheSpot Color Method.

By printing dot patterns(Rosettes) with four colors andallowing the readers eye and perception process to createdthe desired color combinations (Process Color Method).

There is a numbering system for commercial links called ThePantone Matching Systems.


CMYK Color Model

Assigns every commercial ink color a number in accordingwith its percentage of C,M,Y, K.

Describing the ink color as a linear combination of these usingcoefficients between 0 and 1 inclusive.

The set of all such linear combinations is called CMYK-Space.

For example 876CVC is a mixture of C, M, Y, K in theP876 = (0.38, 0.59, 0.73, 0.07).


Linear Independence/Dependence

Definition

Let S = {v1, v2, , vn} be a finite subset of a vector space V .Then S is called linearly independent iff

ni=1 aivi = 0 has a trivial

solutions, i.e., all ai are simultaneously zero.

Definition

Let S = {v1, v2, , vn} be a finite subset of a vector space V .Then S is called linearly dependent iff

ni=1 aivi = 0 has a

nontrivial solutions, i.e., at least one of the ai is nonzero.


Examples

Example (1)

Check whether S = {x2 + x + 1, x2 1, x2 + 1} is L.I. or L.D.???

Example (2)

Check whether S = {(2,5, 1), (1, 1,1), (0, 2,3), (2, 2, 6)} isL.I. or L.D.???


Linear Dependence

Theorem

Let S be a nonempty finite subset of a vector space V . Then S islinearly dependent if and only if for some vector v span(S) canbe expressed as a linear combination of the elements of S.


Remark

A set of vectors S in a vector space V is linearly independent iffthere is no vector v S such that v span(S \ v).

Remark

A set S in a vector space V is linearly dependent iff there is somevector v S such that v span(S \ v).


Linear Dependence

Remark

A nonempty set of vectors S = {v1, v2, , vn} is linearlyindependent iff

v1 6= 0. 2 k n, vk does not belongs to span(S).


Linear Independence

Theorem

Let S be a nonempty finite subset of a vector space V . Then S islinearly independent if and only if every vector v span(S) can beexpressed uniquely as a linear combination of the elements of S.


Theorem

If S is any set in Rn containing k distinct vectors, where k > n,then S is linearly dependent.


Linearly dependent of an infinite set

Definition

An infinite subset S of a vector space V is linearly dependent iffsome finite subset T of S is linearly dependent.

Definition

An infinite subset S of a vector space V is linearly independent ifit is not linearly dependent.


Thank you


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