GOVT.POLYTECHNIC-TRB UNIT-I&III. 10% DISCOUNT FOR ALL POLYTECHNIC TRB MATERIALS

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1 STUDY MATERIAL-CONTACT: GOVT.POLYTECHNIC-TRB MATHEMATICS UNIT-I&III ALGEBRA REAL ANALYSIS 0% DISCOUNT FOR ALL POLYTECHNIC TRB MATERIALS POLYTECHNIC TRB MATERIALS MATHS / ENGLISH COMPUTER SCIENCE / IT / CHEMISTRY / PHYSICS AVAILABLE CONTACT AVAILABLE

2 STUDY MATERIAL-CONTACT: ANNIHILATING POLYNOMIAL: The aihilator of a set S V is Sa = {f V * f (v) = 0, v S}. Defiitio: Let V be a vector space over the field F, where F = (or) F = C. A ier product o V is a fuctio, : V V F such that for all u,v,w V ad a,b F, the followig hold: (i) v,v 0 ad (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 deotes complex cojugatio). A real (or complex) ier product space is a vector spacev over (or C), together with a ier product defied o it. I a ier product space V, the orm, orlegth, of a vector v V is v = v, v. A vectorv V is a uit vector if v =. The agle betwee two ozero vectors u ad v i a real ier product space is the real umber θ, 0 θ π, such that u,v = u v cosθ. Let V be a ier product space. The distace betwee two vectors u ad v is d(u,v)= u v. Gram Schmidt Orthogoalizatio process: Defiitio: Let {a,a,...,a } be a basis for a subspace S of a ier product space V. A orthoormal basis {u,u,...,u }for S ca be costructed usig the followig Gram Schmidt orthogoalizatio process: AVAILABLE

3 STUDY MATERIAL-CONTACT: u a / a ad u = a a, ui ui / a i i a, ui ui for =,..,. Jorda Caoical Form: A Jorda caoical form of matrix A, deoted JA (or) JCF(A), is a Jorda matrix that is similar to A. It is covetioal to group the blocs for the same eige value together ad to order the Jorda blocs with the same eige value i oicreasig size order. Jorda caoical form: Let A C x have the Jorda caoical form Z - AZ = J A = diag (J λ ), J (λ ),..., J p (λ p ),where Z is o sigular, m xm j ( ) C ad m +m + +m p =. The Jorda i variats of A are the followig parameters: The set of distict eigevalues of A. For each eigevalueλ, the umber b λ ad sizes p,..., p b λ of the Jorda blocs with eigevalue λ i a Jorda caoical form of A. Schur Complemets: The Schur complemet of A i A is the matrix A A A - A, sometimes deoted A/A. AVAILABLE

4 STUDY MATERIAL-CONTACT: Direct Sum Decompositios: The sum of subspaces W i, for i =,...,, is i = W i = W + +W ={w + +w w i W i }. The sum W + +W is a direct sum if for all i =,...,, we have W i W j =i Wj ={0}. W = W W deotes that W = W + +W ad the sum is direct. The subspaces W i, for i =i,...,, are idepedet iff (or) w i W i, w + +w =0 implies w i =0 for all i =,...,. Let V i, for i =,...,, be vector spaces over F. Ier Product Spaces: Let V be a real vector space. Suppose to each pair of vectors u,v V there is assiged a real umber, deoted by u, v. This fuctio is called a (real) ier product o V if it satisfies the followig axioms:. (Liear Property): au bu, v a u,v +b b u, v.. (Symmetric Property): u, v = v, u 3. (Positive Defiite Property): u, u 0& u,u 0 iff u=0. The vector space V with a ier product is called a (real) ier product space. Example of Ier Product Spaces:.Let u=(,3,-4,), v =(4,-,,), w =(5,-,-,6) i R 4. Show that <3u-v,w>=3<u,w>-<v,w>. Sol. By defiitio, <u,w>=5-3+8+= ad <v,w>=0+-4+6=4. Note that, 3u-v=(-5,3,-6,4). Thus, <3u-v,w>= =8. AVAILABLE

5 STUDY MATERIAL-CONTACT: As expected,3<u,w>-<v,w>=3()-(4)=8. There fore, 3<u,w>-<v,w>=<3u-v,w>. Cayley Hamilto Theorem: Every matrix A is a root of its characteristic polyomial. (or) Every square matrix satisfies its characteristic equatio. Ivariat Direct-Sum Decompositios: A vector space V is termed the direct sum of subspaces W,..,W r, writte V =W W,... W r if every vector v ɛ V ca be writte uiquely i the form v= w +w +.+w r, with w i ɛ W i. Primary Decompositio Theorem: Let T:V V be a liear operator with miimal polyomial m(t)=f (t) f (t),. f r (t) r where the f i (t) are distict moic irreducible polyomials. The V is the direct sum of T-ivariat subspaces W,...,Wr, where W i is the erel of f i (T) i. Moreover, f i (t) i is the miimal polyomial of the restrictio of T to W i. AVAILABLE

6 STUDY MATERIAL-CONTACT: GOVT.POLYTECHNIC COLLEGE-LECTURER MATHEMATICS UNIT-I REAL ANALYSIS Discotiuities:. First id: (ifiite) : f ( x ) f ( x ) (ie.,) f ( x ) & f ( x ) exists but ot equal (or) ot both fiite.. Secod id: (Jump discotiuity): Either f ( x )( or) f ( x ) does ot exists(or) both fiite. 3. Third id: (Removable or Poit discotiuity) f ( x ) f ( x ) f ( x) (or) lim f ( x) f ( a ) fiite. Example: si x x xa. If f ( x), x 0, f (0) 37, the f has a removable discotiuity at x=0.. If f ( x) 0, x 0, f ( x), x 0, f (0) 75, the f has a Jump discotiuity at x=0. x 3. If f ( x) 0, x 0, f ( x), x 0, the f has a ifiite discotiuity at x=0. Theorem: Let f:r R,ad assume f mootoe. The all discotiuities of f are jumps ad f has at most coutably may discotiuities. AVAILABLE

7 STUDY MATERIAL-CONTACT: REAL ANALYSIS PROBLEMS. Explai why each of the followig sequeces coverges ad i the case of (i) ad (ii) determie the limits Sol. The sequece coverges because it is a combiatio of stadard coverget sequeces. We have / 4 99 / 3/ 03 4 as.. Determie 3 Sol. 3 ( ( 3) 3) 3 ( 3/ ) 3 ( 3/ ) 3 as. x 3. Determie the limits 3. AVAILABLE

8 STUDY MATERIAL-CONTACT: Sol. ( ) is a sequece of partial sums (i.e. a series). With a = (3 + )/ the ratio test gives a a 3( ) 3 ( ) 3( / ) (/ / 3 / ) as. 4. Defie what it meas for the sequece (x) to be a Cauchy sequece? Sol ( ) is a Cauchy sequece if for every > 0 there exists a N such that x m < or all,m satisfyig N ad m N. 5. If a, b, s a ad t b. Fid the limits of the sequeces (a+/a) ad (b+/b). Sol. Let a a ( ) ( ) ( ) ( / ) as b b ( ) ( ) / as. 6. Determie the limits of the followig sequeces (x) whose th term x is give below Sol (/ ) ( / ) (5/ ) (4 / ) 7 4 as. AVAILABLE

9 STUDY MATERIAL-CONTACT: I the above we have used the result that / 0 as ad a result about combiig coverget sequeces ad otig that the deomiator coverges to a o-zero value. 7. Determie Sol. 3 3 ( / ) (/ ) (6 (/ ) (4 / ) 6 as. We have used the result that / 0 as ad a result about combiig coverget sequeces ad otig that the deomiator coverges to a o-zero value. 8. Determie 3 4. (/ ) (/ ) Sol. as (4 / ) 3 We have used the result that / 0 as ad a result about combiig coverget sequeces. The deomiator coverges ad it is at least 3 for all. Exteded Real umber system: Defiitio: I treatig ad - as umbers, we are extedig the real umber system. We have is RU{-, }. This is called the exteded real umber system Note: It is sometimes deoted R #. AVAILABLE

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