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?
Tarkeshar Singh Linear Algebra
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 .
Tarkeshar Singh Linear Algebra
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 .
Tarkeshar Singh Linear Algebra
If W is a collection of all such vectors w V then W is asubspace of the vector space V .
Tarkeshar Singh Linear Algebra
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 .
Tarkeshar Singh Linear Algebra
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 .
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
Span
Corollary
Let S1,S2 be two subset of a vector space V with S1 S2. Thenspan(S1) span(S2).
Tarkeshar Singh Linear Algebra
Simplifying span
Remark
The span of an empty set is {0}.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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).
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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.???
Tarkeshar Singh Linear Algebra
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.???
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
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).
Tarkeshar Singh Linear Algebra
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).
Tarkeshar Singh Linear Algebra
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).
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
Theorem
If S is any set in Rn containing k distinct vectors, where k > n,then S is linearly dependent.
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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.
Tarkeshar Singh Linear Algebra
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.
Tarkeshar Singh Linear Algebra
Thank you
Tarkeshar Singh Linear Algebra