r r Prof. Inder K. Rana Room 112 B Department of Mathematics - - PowerPoint PPT Presentation
r r Prof. Inder K. Rana Room 112 B Department of Mathematics IIT-Bombay, Mumbai-400076 (India) Email: ikr@math.iitb.ac.in Lecture 13 Prof. Inder K. Rana Department of Mathematics, IIT - Bombay Abstract
Room 112 B Department of Mathematics IIT-Bombay, Mumbai-400076 (India) Email: ikr@math.iitb.ac.in Lecture 13
Department of Mathematics, IIT - Bombay
The algebraic structures of addition and scalar multiplication on I R3 generalized to the notion of Vector Spaces that arise in many other branches of mathematics and other disciplines. Definition (Abstract vector space) Let V be a non-empty set and I F “ I Rpor I
1
VECTOR ADDITION: ` : V ˆ V Ý Ñ V, pa, bq ÞÑ a ` b and
2
SCALAR MULTIPLICATION: p¨q : I F ˆ V Ý Ñ V, pλ, aq ÞÑ λa is called a vector space if the following eight axioms hold:
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@ a, b, c P V and λ P Definition (continuation) A1 a ` b “ b ` a. [Commutativity] A2 pa ` bq ` c “ a ` pb ` cq. [Associativity] A3 D! 0 P V s.t. a ` 0 “ a. [Additive identity] A4 D! ´ a s.t. a ` p´aq “ 0. [Additive inverse] M1 λpa ` bq “ λa ` λb. [Distributivity(scal. mult. over vect. add.)] M2 pλ ` µqa “ λa ` µa [Distributivity(scal. mult. over scal. add.)] M3 λpµaq “ pλµqa [mixed associativity] M4 and 1 ¨ a “ a [normalization]
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The following natural properties can be logically deduced form the above axioms. 0 ¨ a “ 0 “ λ ¨ 0. p´1qa “ ´a. Remark: The uniqueness assertion in A3 and A4 is also derivable if we chose to omit it from the axioms. i.e. write D instead of D!
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Example (Vector spaces)
1
I Rn and I Cn.
2
tx P I Rn|Ax “ 0u, A P MmˆnpI Rq.
3
MmˆnpI Rq and MmˆnpI Cq.
4
I R3rxs, the set of all the polynomials in x with real coefficients of degree ď 3. Similarly I C3rxs or I Rdrxs, etc., can be defined.
5
I Rrxs, pI Crxsq, the set of all the polynomials in variable x with real (complex) coefficients.
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Solutions of the equation y2 ` µ2y “ 0 or of the equation y1 ` qpxqy “ 0 (over some interval I).
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The set of vector functions yptq “ „y1ptq y2ptq satisfying : y “ Ay, where A P M2pI Rq.
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Example (Non vector spaces)
1
All m ˆ n real matrices with entries ě 0.
2
Solutions of xy1 ` y “ 3x2.
3
Solutions of y1 ` y2 “ 0. Definition Let V be a vector space. A subset W of V is called a subspace of V if for w1, w2 P W and α, β P I F, αw1 ` βw2 P W.
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Definition (Linear combinations) Given v1, v2, ..., vk P V, a linear combination is a vector c1v1 ` c2v2 ` ¨ ¨ ¨ ` ckvk for any choice of scalars c1, c2, ..., ck. Definition Let S be a nonempty subset of V. Let rSs :“ řn
i“1 αivi | n P I
N; α1, ...αn P I F; v1, ...vn P S ( . It is called the subspace generated by S. Definition Let V be a vector space such that there exists a finite set S Ă V with rSs “ V. In such a case we say that V is a finitely generated vector space.
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Finitely generated Vector spaces
1
I Rn and I Cn,
2
NpAq :“ tx P I Rn|Ax “ 0u, A P MmˆnpI Rq.
3
MmˆnpI Rq and MmˆnpI Cq.
4
I R3rxs -the set of all the polynomials in x with real coefficients of degree ď 3. Similarly I C3rxs or I Rdrxs can be defined. Vector spaces which is not finitely generated I Rrxs -the set of all the polynomials in x with real coefficients. Similarly I Crxs.
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Definition (Linear dependence) A set of vectors v1, v2, ..., vk P V is called linearly dependent If scalars c1, c2, ..., ck, at least one non zero, can be found such that c1v1 ` c2v2 ` ¨ ¨ ¨ ` ckvk “ 0. Contrapositive of linear dependence is linear independence
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Example
1
In I Rrxs the set t1, x, x2, x5u is a linearly independent set: If c1 ` c2 x ` c3 x2 ` c4 x5 “ 0, right hand side being the zero polynomial, implies, by equating coefficients of like powers, c1 “ c2 “ c3 “ c4 “ 0.
2
On the other hand the set t1, x, 1 ` x2, 1 ´ x2u Ă I Rrxs is a linearly dependent set since p´2q ˆ 1 ` 0 ˆ x ` 1 ˆ p1 ` x2q ` 1 ˆ p1 ´ x2q “ 0, implying there is a linear combination which is zero but not all the scalers are zero.
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Definition (Basis) If a set of vectors S in a vector space V is such that S is linearly independent and Every vector in V is a (finite) linear combination of vectors from S. then the set S is called a basis of S. Theorem (Existence of basis)
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Every vector space V has a basis.
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If V is finitely generated, then any two basis will have same number of elements. The number of elements in any basis of V is called the dimension
3
If V is not finitely generated, we say it is infinite dimensional.
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Example
1
A basis of I R5rxs is S “ t1, x, x2, x3, x4, x5u. For this note that any polynomial p P I R5rxs is ppxq “ c0 ` c1x1 ` c2x2 ` c3x3 ` c4x4 ` c5x5 ` c6x6 and c0 ` c1x1 ` c2x2 ` c3x3 ` c4x4 ` c5x5 ` c6x6 “ 0 implies each ci “ 0. Hence S is a basis of I R5rxs, which says that it is 6-dimensional.
2
is Consider S “ t1, x, x2, ..., xn, ...u Ă I
S is a linearly independent set. Hence I Rrxs is infinite
Rrxs.
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Example ti, j, ku is the standard basis of I R3. te1, e2, ..., enu is the standard basis of I Rn. tEjk, 1 ď j ď m, 1 ď k ď nu Ă MmˆnpI Rqu is a basis of MmˆnpI Rq. tcos µx, sin µxu is a basis of the solution space of the differential equation y2 ` µ2x “ 0. Let V “ tppxq P I R3rxs ˇ ˇpp1q “ 0u. Then V is a vector space. It has B “ t1 ´ x, 1 ´ x2, 1 ´ x3u as a basis. It has 3 dimensions.
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Let Let B :“ tv1, . . . , vnu be any ordered basis of a vector space V over I F of dimension n.For v P V, if α1, α2, . . . , αn P I R are the unique scalars such that v “ α1v1 ` α2v2 ` . . . ` αnvn, then we write rvsB :“ » — – α1 . . . αn fi ffi fl and call it the coordinate vector of v. This gives us an identification v ÞÑ rvsB from V to I Fn. It is easy to show that (i) ru1 ` u2sB “ ru1sB ` ru2sB. (ii) rλusB “ λrusB. Thus for all practical purposes, V – I Fn.
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Definition (Linear map or transformation) Let V, W be two (abstract) vector spaces. A map T : V Ý Ñ W is called a linear map or transformation if Tpv ` wq “ Tv ` Tw and Tpλvq “ λTv. Example A P MmˆnpI Rq can be viewed as a linear map A : I Rn Ý Ñ I Rm via the matrix multiplication v ÞÑ Av. Here v is treated as n ˆ 1 column which maps to m ˆ 1 column Av.
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(a) d dx : I Rrxs Ý Ñ I Rrxs (b) τc : I Rrxsm Ý Ñ I Rrxsm where τc ` ppxq ˘ “ ppx ´ cq for any scalar c P I R. (c) µx : I Rrxsm Ý Ñ I Rrxs defined by µx ` ppxq ˘ “ xppxq. (d) I : Cpr0, 1sq Ý Ñ Cpr0, 1sq defined by ` Ipfq ˘ pxq “ ż x fptqdt, 0 ď x ď 1. Levels of complexity of maps:
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Constant maps. Easiest to describe. Need to know at any one point only.
2
Linear maps. Easiest among non constant maps. Enough to sample on a basis to determine it completely. (Constant maps can not describe dependence of output on input.)
3
Non-linear maps. Hardest to deal with. Often the local behaviour is studied by linear approximation. Hence the importance of linear maps.
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Let T : V Ý Ñ W is a linear map, where V and W are finite dimensional vector spaces with ordered bases B1 “ tv1, v2, ..., vnu and B2 “ tw1, w2, ..., wmu of V and W,respectively. Let Tv1 “ t11w1 ` t21w2 ` ¨ ¨ ¨ ` tm1wm Tv2 “ t12w1 ` t22w2 ` ¨ ¨ ¨ ` tm2wm . . . . . . . . . Tvn “ t1nw1 ` t2nw2 ` ¨ ¨ ¨ ` tmnwm This gives us a (unique) matrix representation rtjks of T w.r.t. to the
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Conversely if A “ rajks is any m ˆ n matrix of scalars, then we can construct a unique linear map TA : V Ý Ñ W by setting TApvq “ TAp ÿ
k
ckvkq linear “ ÿ
k
ckTAvk “ ÿ
j,k
ckajkwj. This can also be written as » — — — – a11 a12 ¨ ¨ ¨ a1n a21 a22 ¨ ¨ ¨ a2n . . . . . . . . . am1 am2 ¨ ¨ ¨ amn fi ffi ffi ffi fl » — — — – c1 c2 . . . cn fi ffi ffi ffi fl “ » — — — – b1 b2 . . . bm fi ffi ffi ffi fl We can describe T as T ´
n
ÿ
k“1
ckvk ¯ “
m
ÿ
j“1
bjwj “ ÿ
j
´ ÿ
k
ajkck ¯ wj.
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Example Consider the linear map τ1 : I R3rxs Ý Ñ I R3rxs as described below. We want to describe its matrix relative to the ordered basis B “ t1, x, x2, x3u on both the sides. Since τ1ppxq “ ppx ´ 1q. Hence
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τp1q “ 1 “ 1.1 ` 0.x ` 0.x2 ` 0.x3.
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τpxq “ px ´ 1q “ p´1q.1 ` 1.x ` 0.x2 ` 0.x3.
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τ1px2q “ px ´ 1q2 “ 1.1 ` p´2q.x ` 1.x2 ` 0.x3.
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τ1px3q “ px ´ 1q3 “ p´1q.1 ` 3x ` p´3qx2 ` 1.x3. Therefore the 4 ˆ 4 matrix of τ1 w.r.t. the basis B is » — — – 1 ´1 1 ´1 1 ´2 3 1 ´3 1 fi ffi ffi fl
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Let T : V Ý Ñ W be a linear transformation. Definition (Null space) The set tv P V ˇ ˇTv “ 0u is called the null space of T. It is also called the kernel of T. It is a vector subspace of V and usually denoted as NpTq. Definition (Range space) The set tTv P W ˇ ˇv P Vu is known as the range space of T. The range of T is denoted by RpTq and it is a vector subspace of W.
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Definition (Rank and nullity) The dimension of NpTq is called the nullity of T and is denoted νpTq. The dimension of RpTq is called the rank of T and is denoted ρpTq. Theorem (Rank-nullity theorem) For any linear transformation T : V Ý Ñ W between two finite dimensional vector spaces we have ρpTq ` νpTq “ dimV.
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