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ANNIHILATING POLYNOMIAL:
The annihilator of a set S V is Sa = {f V* | f (v) = 0, v S}.
Definition:
Let V be a vector space over the field F, where F = (or) F = C. An inner
product on V is a function , : V V F such that for all u,v,w V and a,b
F, the following hold:
(i) v,v 0 and (v,v =0 iff v=0.
(ii) au+bv,w = a u,w +b v,w .
(iii) For F = : u,v = v,u ;
For F = C: u,v = v,u (where bar denotes complex conjugation).
A real (or complex) inner product space is a vector spaceV over (or
C), together with an inner product defined on it. In an inner product space V, the
norm, orlength, of a vector v V is ||v||= vv, . A vectorv V is a unit vector if ||v||=1.
The angle between two nonzero vectors u and v in a real inner product
space is the real number , 0 , such that u,v =||u|| ||v|cos.
Let V be an inner product space. The distance between two vectors u and v
is d(u,v)=||uv||.
GramSchmidt Orthogonalization process:
Definition:
Let {a1,a2,...,an} be a basis for a subspace S of an inner product space V. An
orthonormal basis {u1,u2,...,un}for S can be constructed using the following Gram
Schmidt orthogonalization process:
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||||/ 111 aau and uk =
1
1
,
k
i
iikk uuaa /
1
1
, ||||k
i
iikk uuaa for k=2,..,n.
Jordan Canonical Form:
A Jordan canonical form of matrix A, denoted JA (or) JCF(A), is a
Jordan matrix that is similar to A. It is conventional to group the blocks for the same
eigen value together and to order the Jordan blocks with the same eigen value in non-
increasing size order.
Jordan canonical form:
Let A Cnxn have the Jordan canonical form Z-1 AZ = JA = diag
(J1 1 ), J2 (2 ),..., Jp (p ),where Z is non singular,
kk xmm
k
k
k
kk Cj
1
1
)(
and m1+m2 ++mp =n.
The Jordan in variants of A are the following parameters:
The set of distinct eigenvalues of A.
For each eigenvalue, the number b and sizes p1,..., pb of the Jordan blocks
with eigenvalue in a Jordan canonical form of A.
Schur Complements:
The Schur complement of A11 in A is the matrix A22 A21 A11-1 A12 ,
sometimes denoted A/A11.
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Direct Sum Decompositions:
The sum of subspaces Wi, for i = 1,...,k, is k i=1
Wi= W1 ++Wk={w1 ++wk |wi Wi}.
The sum W1 ++Wk is a direct sum if for all i = 1,...,k, we have Wi Wj =i
Wj ={0}. W = W1 Wk denotes that W = W1 ++Wk and the sum is direct.
The subspaces Wi, for i =i,...,k, are independent iff (or) wi Wi, w1++wk =0
implies wi =0 for all i =1,...,k. Let Vi, for i =1,...,k, be vector spaces over F.
Inner Product Spaces:
Let V be a real vector space. Suppose to each pair of vectors u,v V there is
assigned a real number, denoted by vu, . This function is called a (real) inner product
on V if it satisfies the following axioms:
1. (Linear Property): vbuau ,21 v,u1a +b v,u 2b .
2. (Symmetric Property): vu, = u, v
3. (Positive Definite Property): 0u u, & 0u,u iff u=0.
The vector space V with an inner product is called a (real) inner product space.
Example of Inner Product Spaces:
1.Let u=( 1,3,-4,2), v =(4,-2,2,1), w =(5,-1,-2,6) in R4. Show that
=3-2.
Sol.
By definition, =5-3+8+12=22 and =20+2-4+6=24.
Note that, 3u-2v=(-5,13,-16,4).
Thus, =-25-13+32+24=18.
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As expected,3-2=3(22)-2(24)=18.
There fore, 3-2=.
CayleyHamilton Theorem:
Every matrix A is a root of its characteristic polynomial.
(or)
Every square matrix satisfies its characteristic equation.
Invariant Direct-Sum Decompositions:
A vector space V is termed the direct sum of subspaces W1,..,Wr,
written V =W1W2 ,.....Wr if every vector v V can be written uniquely in the
form v= w1 +w2+.+wr, with wi Wi.
Primary Decomposition Theorem:
Let T:V1 V be a linear operator with minimal polynomial m(t)=f1
(t)n1 f2(t)n2,. fr(t)
nr where the fi(t) are distinct monic irreducible polynomials.
Then V is the direct sum of T-invariant subspaces W1,...,Wr, where
Wi is the kernel of fi (T)ni. Moreover, fi (t)
ni is the minimal polynomial of the
restriction of T to Wi.
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GOVT.POLYTECHNIC COLLEGE-LECTURER
MATHEMATICS
UNIT-I
REAL ANALYSIS
Discontinuities:
1. First kind: (infinite) :
)()( xfxf (ie.,) )(&)( xfxf exists but not equal (or) not both finite.
2. Second kind: (Jump discontinuity):
Either )())(( xforxf does not exists(or) both finite.
3. Third kind: (Removable or Point discontinuity)
)()()( xfxfxf (or) )()(lim afxfax
finite.
Example:
1. If ,37)0(,0,sin
)( fxx
xxf then f has a removable discontinuity at x=0.
2. If ,175)0(,0,1)(,0,0)( fxxfxxf then f has a Jump discontinuity at x=0.
3. If ,0,1
)(,0,0)( xx
xfxxf then f has an infinite discontinuity at x=0.
Theorem:
Let f:RR ,and assume f monotone. Then all discontinuities of f are jumps
and f has at most countably many discontinuities.
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REAL ANALYSIS PROBLEMS
1. Explain why each of the following sequences converges and in the case of (i)
and (ii) determine the limits 3994
81032
2
nn
nxn .
Sol. The sequence converges because it is a combination of standard
convergent sequences.
We have 3994
81032
2
nn
nxn
2
2
/3/994
/8

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