MATHÉMATIQUES : PROGRAMMES LYCÉE SCOLARIA ...

E Ii Section

Description

• cours - matière potentielle : specifique
• cours - matière potentielle : mathematiques
• cours - matière potentielle : logique mathematique

MATHEMATIQUES : PROGRAMMES LYCEE SCOLARIA EXCELLENCE INTRODUCTION Les programmes Scolaria de mathematiques au lycee prennent pour point de depart les programmes de l'Education Nationale, afin d'assurer l'objectif de 100% de reussite au baccalaureat pour nos eleves (bac S en filiere scientifique, bac L en filiere linguistique). Les textes de programmes qui suivent se composent grosso modo de la partie ”contenus” du programme de l'EN (la plupart des commentaires et indications de mise en oeuvre ont ete supprimes) auxquels s'ajoutent quelques modifications de notre cru.

• idem
• rappel de la definition de la limite
• −−→ ab
• seconde relais
• calcul des coordonnees du point d'intersection
• programmes de terminale scientifique
• definition
• vecteurs −−→
• vecteur −−→
• vecteur
• vecteurs
• translation

Sujets

• mathématiques
• mathématique
• Mathématiques

Informations

• Publié par: epsa0
• Nombre de lectures: 56

Extrait

Revised Curriculum

MATHEMATICS
(UG courses)
III YEAR Degree - Effective from the academic year 2010-11






















JANUARY 2010
S.V.UNIVERSITY

















SRI VENKATESWARA UNIVERSITY : TIRUPATI

MMOODDEELL CCUURRRRIICCUULLUUMM -- B.A/B.Sc B.A/B.Sc

90 hrs Mathematics: Paper – III
(3 hrs/ week) LINEAR ALGEBRA AND VECTOR CALCULUS
Part A: Linear Algebra
UUnniitt--II:: ((2200 HHoouurrss))
Vector spaces, General properties of vector spaces, Vector subspaces, Algebra of subspaces, lineeaarr
combination of vectors. Linear span, linear sum of two subspaces, Linear independence and dependence ce
of vectors, Basis of vector space, Finite dimensional vector spaces, Dimension of a vector space, ce,
Dimension of a subspace.

UUnniitt –– IIII ((1155 HHoouurrss))
Linear transformations, linear operators, Range and null space of linear transformation, Rank and nullity Linear transformations, linear operators, Range and null space of linear transformation, Rank and nullity
of linear transformations, Linear transformations as vectors, Product of linear transformations, Matrix of of linear transformations, Linear transformations as vectors, Product of linear transformations, Matrix of
Linear transformation

Unit-III: (25 Hours)
MMaattrriicceess,, EElleemmeennttaarryy pprrooppeerrttiieess ooff mmaattrriicceess,, IInnvveerrssee mmaattrriicceess,, RRaannkk ooff mmaattrriixx,, LLiinneeaarr eeqquuaattiioonnss
characteristic values & vectors, Cayley – Hamilton theorem characteristic values & vectors, Cayley – Hamilton theorem
Inner product spaces, Euclidean and unitary spaces, Norm or length of a vector, Schwartz inequality, ity,
Orthogonality, Orthonormal set, complete orthonormal set, Gram - Schmidt orthogonalisation processss..
Bessel’s inequality

Prescribed text book: Linear Algebra by J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir, Prescribed text book: Linear Algebra by J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir,
Meerut-250002. Matrices by Shanti Narayana (S.Chand Publications) Meerut-250002. Matrices by Shanti Narayana (S.Chand Publications)
RReeffeerreennccee BBooookkss:: 11.. LLiinneeaarr AAllggeebbrraa bbyy KKeennnneetthh HHooffffmmaann aanndd RRaayy KKuunnzzee,, PPeeaarrssoonn EEdduuccaattiioonn ((llooww
priced edition), New Delhi
th2. Linear Algebra by Stephen H. Friedberg et al Prentice Hall of India Pvt. Ltd. 4 edition 2007

Part B : Vector Calculus
Unit-IV: Vector differentiation ( 15 Hours) Unit-IV: Vector differentiation ( 15 Hours)
VVeeccttoorr ddiiffffeerreennttiiaattiioonn.. OOrrddiinnaarryy ddeerriivvaattiivveess ooff vveeccttoorrss,, DDiiffffeerreennttiiaabbiilliittyy,, GGrraaddiieenntt,, DDiivveerrggeennccee,, CCuurrll
operators, Formulae involving these operators.
Prescribed book: A Course of Mathematical Analysis by Santhi Narayana and P.K.Mittal, S. Chand d
Publications..

UUnniitt--VV:: VVeeccttoorr IInntteeggrraattiioonn (( 2200 HHoouurrss))
LLiinnee iinntteeggrraall,, ssuurrffaaccee iinntteeggrraall,, vvoolluummee iinntteeggrraall wwiitthh eexxaammpplleess
Vector integration, Theorems of Gauss and Stokes, Green’s theorem in plane and applications of these Vector integration, Theorems of Gauss and Stokes, Green’s theorem in plane and applications of these
theorems.

Prescribed text Book: A Course of Mathematical Analysis by Santhi Narayana and P.K.Mittal, S.
Chand Publications..

Reference Books:
1.Text book of vector Analysis by Shanti Narayana and P. K. Mittal,
S. Chand & Company Ltd, New Delhi.
2. Mathematical Analysis by S.C. Mallik and Savitha Arora, Wiley Eastern Ltd.
3. Vector Analysis by Murray. R.Spiegel, Schaum series Publishing Company ,
III B.A / B.Sc Mathematics
PAPER – III PAPER – III
(Linear Algebra and Vector Calculus) (Linear Algebra and Vector Calculus)
(Revised from 2010-2011)
QUESTION BANK FOR PRACTICAL

UNIT-I UNIT-I
1. Let V be the set of all pairs (x , y) of real numbers and let F be the field of real
(,xyx) +( ,y )=+(xxy+y )11 1, 1
numbers.Define
cx(,y)=−(cx, y).
Show that with these operations V is not a vector space over the field of real numbers. Show that with these operations V is not a vector space over the field of real numbers.
2. Prove that the set of all solutions (a, b, c) of the equation a+b+2c = 0 is a
subspace of the vector space V ( R ). 3
3. Show that the set W of the elements of the vector space V ( R ) of the form 3
(x+2y, y, -x+3y) where x ,yR∈ is a subspace of V ( R ). 3
4. If a vector space V is the set of all real valued continuous functions over the field 4. If a vector space V is the set of all real valued continuous functions over the field
of real number R, then show that the set W of solutions of the differential equation
2dy dy
29−+2y=0 is a subspace of V 2dx dx
5. Show that the system of vectors (1,3,1), (1,-7,-8), (2,1,-1) of V ( R ) is linearly 3
dependent dependent
66.. SShhooww tthhaatt tthhee ssyysstteemm ooff vveeccttoorrss ((11,,22,,00)),, ((00,,33,,11)),, ((--11,,00,,11)) ooff VV (( QQ )) iiss LL..II wwhheerree QQ 33
is the field of rational numbers.
7. if α,,βγ are linearly independent vectors of V( R ) show that α++ββ,, γ γ+α arree
also L.I
S ={,α βγ, ,δ}88.. TThhee sseett wwhheerree α = (1, 0, 0),βγ==(1,1, 0), (1,1,1)andδ= (0,1, 0) iiss aa ssppaacciinngg 4
3 3 set of R ( R) but not a basis of set.
9. (i) Show that the set {(1,2,1),(2,1,0),(1,-1,2)} forms a basis of V ( F ). 3
3(ii) Show that the set of vectors {(2,1,4),(1,-1,2),(3,1,-2)} form a basis of R
10. If W is the subspace of V ( R ) generated by the vectors (1,-2,5,-3),(2,3,1,-10. If W is the subspace of V ( R ) generated by the vectors (1,-2,5,-3),(2,3,1,-44
4) and (3,8,-3,-5) find a basis of W and its dimension
11. V is the space generated by the polynomials
32 3 2 3 2αβ=+xx22−x+1, =+x 3x −x+4,γ=2x +x +−7x−7 . Find a basis of V . Find a basis of V
and its dimension. and its dimension.
412. Let W and W be two subspaces of R given by W = { (a,b,c,d):b-1 2 1
2c+d=0}, W = { (a,b,c,d):a=d,b=2c}.Find the basis and dimension of 2
()i W (ii) W (iii)W ∩W and hence find dim(W + W ) and hence find dim(W + W ) 11 2212 1 2
UNIT-II UNIT-II
1. The mapping TV:(R) →V(R) is defined by T(x,y,z)= (x-y,x-z). Showingg 32
that T is a linear transformation.
22 22. The mapping TV:(R) →V(R) is defined by Ta (,b,c)=++a b c Can T be a 2. The mapping is defined by Can T be a 31
linear transformation?
23. Describe explicitly the linear transformation TR: →R such that T 3. Describe explicitly the linear transformation such that T
(2,3)=(4,5) and T(1,0) =(0,0).
34. Find T (x,y,z) Where TR: →R is defined by T (1,1,1) = 3, T(0,1,-2)=1, TT
((00,,00,,11))== --22..
225. Find a linear transformation , such that T (1,0) =(1,1) and TR: →R
T(0,1)=(-1,2). Prove that T map[s the square with vertices (0,0), (1,0), T(0,1)=(-1,2). Prove that T map[s the square with vertices (0,0), (1,0),
(1,1), (0,1) into a parallelogram.
6. let GV : →V and HV : →V be two linear operators defined by 33 33
Ge()=+e eG(e)=e,G(e)=e −e aanndd H(e) =eH,(e)=−2e eH,(e)=0 11 2, 2 3 3 2 3 13 2 2 3 3
Where {,ee,e} is the standard basis of V ( R ) . Find (i) G+H (ii) 2G 312 3
32 327. Let and be defined by T (x,y,z)=(3x,y+z) and 7. Let TR: →R and H :RR→ be defined by T (x,y,z)=(3x,y+z) and
H(x,y,z)=(2x-x,y). Compute (i) T+H (ii) 4T-5H (iii) TH (iv) H T
8. If TV:(R) →V(R) is a linear transformation defined by T(a,b,c,d)=(a-43
bb++cc++dd,,aa++22cc--dd,,aa++bb++33cc--33dd)) ffoorr ab,,c,d ∈R tthheenn vveerriiffyy RRaannkk –– NNuulllliittyy
Theorem 339. Find a linear transformation TR: →R whose range is spanned by
(1,2,0,-4),(2,0,-1,-3). (1,2,0,-4),(2,0,-1,-3).
10. Find the null space, range, rank and nullity of the transformation
33TR: →R ddeeffiinnee bbyy TT((xx,,yy))==((xx++yy,,xx--yy,,yy))..
11. Let TV : →V be defined by T(x,y) = (x+y, 2x-y, 7y) Find [:TB,B] 23 12
where BandB are the standard bases of V and V 12 23.
3212. Let TR: →R be the linear transformation defined by (x , y, z ) = 12. Let be the linear transformation defined by (x , y, z ) =
(3x+2y-4x, x-5y+3x). Find the matrix of T relative to the bases B = 1
B{{((11,,11,,11)),, ((11,,11,,00)),, ((11,,00,,00))}},, =={{((11,,33)),, ((22,,55))}} 2
UNIT-III
1. Reduce the following matrix into normal form and hence find the rank off
0 2 422
444 8 0the matrix. A = 82 0 10 2
63 2 9 1
12 3 0 
 2432 22.. RReedduuccee tthhee mmaattrriixx A = iinn ttoo EEcchheelloonn ffoorrmm aanndd hheennccee ffiinndd iittss
 32 1 3
 
68 7 5 
rank.
024 
 3. Determine whether the matrix A = 242 is invertible. If so find its  
 33 1 
inverse.
44.. SSoollvvee xx++yy--zz++tt == 00 ;; xx--yy++22zz--tt == 00,, 33xx++yy++tt == 00
5. Solve th