Paper III - Curriculum. III B.A./B.Sc. Mathematics PART A : LINEAR ALGEBRA

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1 AHARYA NAGARJUNA UNIVERSITY URRIULUM BA / BSc MATHEMATIS - PAPER - III LINEAR ALGEBRA AND VET ALULUS (Sllabus for the academic ears 00-0 and onwards) Paper III - urriculum 90 Hours PART A : LINEAR ALGEBRA UNIT - I (0 Hours) Vector spaces, General properties of vector spaces, Vector subspaces, Algebra of subspaces, linear combination of vectors Linear span, linear sum of two subspaces, Linear independence and dependence of vectors, Basis of vector space, Finite dimensional vector spaces, Dimension of a vector space, Dimension of a subspace UNIT - II (5 Hours) Linear transformations, linear operators, Range and nullspace of transformation, Rank and nullit of linear transformations, Linear transformations as vectors, Product of linear transformations, Invertable linear transformation Inner product spaces, Euclidean and unitar spaces, Norm or length of a vector, Schwartz inequalit, Orthogonalit, Orthonormal set, complete orthonormal set, Gram - Schmidt orthogonalisation process Pescribed text book : Linear Algebra b JN Sharma and AR Vasishtha, Krishna Prakashan Mandir, Meerut Reference Books : Linear Algebra b Kenneth Hoffman and Ra Kunze, Pearson Education (low priced edition), New Delhi Linear Algebra b Stephen H Friedberg, Arnold J Insal and Lawrance E Spence, Prentice Hall of India Pvt Ltd 4th edition -004 PART B : MULTIPLE INTEGRALS AND VET ALULUS UNIT - III (0 Hours) Vector differentiation Ordinar derivatives of vectors, Space curves, ontinuit, Differentiabilit, Gradient, Divergence, url operators, Formulae involving these operators Pescribed test Book : Vector Analsis b Murra R Spiegel, Schaum series Publishing compan, Related topics in hapters and 4 Reference Books : Text book of vector Analsis b Shanti Naraana and PK Mittal, Shand & ompan Ltd, New Delhi Mathematical Analsis b S Mallik and Savitha Arora, Wile Eastern Ltd UNIT - IV (5 Hours) Multiple integrals : Introduction, the concept of a plane curve, line integral - Sufficient condition for the existence of the integral The area of a subset of R, alculation of double integrals, Jordan curve, Area, hange of the order of integration, Double integral as a limit, hange of variable in a double integration Vector integration, Theorems of Gauss and Stokes, Green's theorem in plane and applications of these theorems Pescribed test Book : A ourse of Mathematical Analsis b Santhi Naraana and PK Mittal, S hand Publications Related topics in hapter - 6 Vector Analsis b Murra R Spiegel, Schaum series Publishing compan Related topics in hapters - 5 and 6 Reference Books : Text book of vector Analsis b Shanti Naraana and PK Mittal, Shand & ompan Ltd, New Delhi Mathematical Analsis b S Mallik and Savitha Arora, Wile Eastern Ltd c c c

2 AHARYA NAGARJUNA UNIVERSITY BA / BSc MATHEMATIS - PAPER - III LINEAR ALGEBRA AND VET ALULUS QUESTION BANK F PRATIALS Paper III - urriculum UNIT - I (VET SPAES) Show that a field K can be regarded as a vector space over an subfield F of K Show that the set M of all m nmatrices with their elements as real numbers is a vector space over the field F of real numbers with respect to addition of matrices as addition of vectors and multiplication of a matrix b a scalar as scalar multiplication Show that the set V n of all ordered n-tuples over a field F is a vector space wrt addition of n-tuples as addition of vectors and multiplication of an n-tuple b a scalar as scalar multiplication 4 Let S be a nonempt set and F be a field Let V be the set of all functions from S into F If addition and scalar multiplication are defined as ( f + g)( x) = f ( x) + g( x) x S and ( cf )( x) = cf ( x) x S where f, g V, c F, then show that V is a vector space 5 If F is a field then show that W = {( a, a, 0 ): a, a F} is a subspace of V( F) 6 Let F be a field and a, a, a be three fixed elements of F Show that W = {( x, x, x): x, x, x F and ax+ ax + ax = 0} is a subspace of V( F) 7 i) If S and T are two subsets of a vector space V(F) then show that S T L( S) L( T) ii) If S is a subset of a vector space V(F) then show that LLS ( ( )) = LS ( ) 8 If S and T are two subsets of a vector space V(F) then show that LS ( T) = LS ( ) + LT ( ) 9 Express the vector α = (,, 5) as a linear combination of the vectors α = (,,), α = (,,), α = (,,) 0 If α = (,,), β = (,, 5), γ = (, 4, 7) then show that the subspaces of V (R) spanned b S = { αβ, } and T = { αβγ,, } are the same Show that the sstem of three vectors (,, ),(, 7, 8),(,, ) of V (R) is linearl dependent Show that ( 4,,, ),(,, 5, ),(, 40,, ),( 6,,, ) are linearl dependent in V 4 (R) Show that {( 0,, ),( 0,, ),( 0,, )} are linearl independent in V (R) 4 Show that the vectors (,,),( 0,, ),(,, ) form a basis for R 5 Show that {(,, 0),(,,),(,,)} form a basis of V over reals and express the vector (,,) as a linear combination of these basis vectors 6 Show that (,,,),(,,,),(, 0 0 0,,),(, 0 0, 0,) is a basis of V 4 over reals Expressα = ( 4,,, ) as a linear combination of these basis vectors 7 Extend the set {( 0,,, ),(, 0,, )} to form a basis for V4 ( R) 8 Find a basis for the subspace spanned b the vectors ( 0,, ),( 0,, ),( 0,, ) in V (R) 9 If W and W are subspaces of V4 ( R)generated b the sets {(,,,), (,,,0), (,,, )} and {(,, 0, ),(,, 0, )} respectivel then find dim ( W + W ) and obtain a basis for W + W 0 W = {( x,, 00, ): x, R} Write a basis for the quotient space R 4 / W

3 Paper III - urriculum UNIT - II (LINEAR TRANSFMATIONS AND INNER PRODUT SPAES) Show that the mapping T : V (R) V (R) defined as Ta (, a, a ) = ( a a + a, a a a ) is a linear transformation from V (R) into V (R) Find a linear transformation TR : R such that T( 0, ) = (,) and T( 0, ) = (, ) Prove that T maps the square with vertices (0, 0), (, 0), (, ), (0, ) into a parallelogram Let T be a linear transformation from a vector space U( F) into a vector space V( F) i) If W is a subspace of U then show that TW ( ) is a subspace of V ii) If X is a subspace of V then show that { α UT : ( α) X } is a subspace of U 4 If W is a subspace of a vector space V and if a linear transformation TV : V/ W is defined b T( x) = W +α for all α V, find the kernel of T 5 Show that the mapping TR : Rdefined b T( x,, z) = ( x+ z, + z, x+ z) is a linear transformation Find its rank, nullit and verif rank T + nullit T = dim R 6 Show that the mapping T : V (R) V (R) defined as Tab (, ) = ( a+ ba, bb, ) is a linear transformation from V (R) into V (R) Find the range, rank, null space and nullit of T 7 If T : V 4 (R) V (R) is a linear transformation defined b Tabcd (,,, ) = ( a b+ c+ d, a+ c d, a+ b+ c d)for a, bcd,, Rthen verif rank T + nullit T = dim V 4 (R) 8 If T : V (R) V (R) is a linear transformation defined b Tabc (,, ) = ( aa, b, a+ b+ c) then show that ( T I)( T I) = 0$ 9 Let T: R R be a linear transformation defined b T( x, x, x) = ( x, x x, x + x + x) Is T invertible? If so find a rule for T like the one which defined T 0 Let T : V (R) V (R) be defined b T x, x, x ) = (x + x, x + x, x + x + 4 ) Prove that T is invertible and find T ( x If α = ( a, a,, an), β = ( b, b,, bn) Vn( ) then show that ( α, β ) = a b + a b + + a b n n defines an inner product on V n () If α = ( a, a), β = ( b, b) V ( R) then show that ( αβ, ) = ab ab ab + 4 ab defines an inner product on V (R) Let V( )be the vector space of all continuous complex valued functions on [0, ] If f (), t g() t V then show that ( f ( t), g( t)) = f ( t) g( t) dt 0 defines an inner product on V( ) 4 Find the unit vector corresponding to ( i, + i, + i ) of V( )with respect to the standard inner product 5 Show that the absolute value of the cosine of an angle cannot be greater than 6 Let V( R)be the vector space of polnomials with inner product defined b ( f, g) = f ( t) g( t) dt for f, g V If f ( x) = x + x 4, g( x) = x x [ 0, ] then find ( f, g), f and g 0

4 4 Paper III - urriculum 7 Show that {(, 00, ),(,, 00),(,,)} 00 is an orthonormal set in the inner product space V( R) 8 Appl the Gram-Schmidt process to the vectors β = (, 0,), β = (, 0, ), β = ( 04,, ), to obtain an orthonormal basis for V( R) with the standard inner product 9 Appl the Gram-Schmidt process to the vectors{(,,),(,,),(,,)} to obtain an orthonormal basis for V( R) with the standard inner product 40 If W and W are two subspaces of a finite dimensional inner product space, prove that i) ( W + W ) = W W ii) ( W W ) = W + W UNIT - III (VET DIFFERENTIATION) dr d r dr d r 4 If r = e i + log( t + ) j tant k then find,,, dt dt dt dt at t = 0 4 If A = 5t i + tj t k and B= sin ti cost j then find i) d dt ( AB ) ii) d dt ( A B) dr d r dr d r d r 4 If r = acosti+ asin t j+ attanθ k then find dt dt and dt dt dt 44 If A= sin ti cost j+ t k, B= costi sin t j k and = i+ j k then find d dt [ A ( B )] at t = 0 45 If f= cos xi+ ( x x ) j ( x + ) k then find f f f,, x x 46 If f= x i z+ xz kand ϕ = z x then find f x ϕ + ϕ + ϕ i j k x z and f ϕ ϕ ϕ i + j + k z x z at (,,) 47 If a = x+ + z, b= x + + z, c= x+ z+ zx then show that [ a b c] = 0 48 Find the directional derivative of ϕ = xz+ 4xzat the point (,, ) in the direction of i j k 49 Find the maximum value of the directional derivative of ϕ = x z 4 at (,, ) 50 Find the directional derivative of f = x + z at P(,, ) in the direction of PQ where Q = (, 504, ) 5 Find the directional derivative of ϕ = x + z along the tangent to the curve x = t, = t, z = t (,,) at 5 Find the angle between the surfaces of the spheres x + + z = 9, x + + z + 4x 6 8z 47= 0at the point ( 4,, )

5 5 Paper III - urriculum 5 Find the angle between the surfaces x + + z = 9 and x + z = at (,, ) 54 Show that r n r is irrotational Find when it is solenoidal 55 Show that i) div r = ii) curl r = 0 iii) grad ( r a) = a iv) div ( r a) =0 56 i) Find p if ( x+ ) i+ ( z) j+ ( x+ pz) k is solenoidal ii) Find the constants abc,, so that ( x + + az) i+ ( bx z) j+ ( 4x + c + z) k is irrotational 57 If F = grad( x + + z xz) then find divf, curlf 58 If F= xz i+ 4x j x z k the find ( F ) and ( F ) at (,, ) Also verif that ( F) = ( F) + F a r a r 59 If a is a constant vector then show that url = + 5 ( ar ) r r r 60 Show that ( r ) = n( n+ ) r n n UNIT - IV (VET INTEGRATION AND MULTIPLE INTEGRALS) 6 Evaluate ( x dx d), where is the curve in the x - plane, = x from (0, 0) to (, ) 6 Evaluate ( x dx + z d + zx dz), where is x = t, = t, z = t, t varing from to 6 Evaluate ( x + ) dx + ( 4x) d around the triangle AB whose vertices are A= ( 00, ), B = ( 0, ), = (, ) over [ 0, ; 0, ] 64 Evaluate x( x + ) dx d 65 Evaluate x( x + ) dx d a b over the area between x = and = x 66 Evaluate ( x + ) dx d over the area bounded b the ellipse x a + = b 67 hange the order of integration in the double integral 0 x e dx d and hence find the value 68 Evaluate ( ab bx a ) dx d the field of integration being the positive quadrant of the ( ab + bx + a ) ellipse x / a + / b = x+ 69 Using the transformation x+ = u, x = v, evaluate e dxd x = 0, = 0, x+ = 70 Evaluate the integral 4a 0 / 4a x x + x dx d b changing to polar coordinates over the region bounded b

6 Paper III - urriculum 7 If F= xi j, evaluate F drwhere is the curve = x in the x -plane from (0, 0) to (, ) 7 If F= ( x + ) i xj, evaluate F dr where the curve is the rectangle in the x -plane bounded b = 0, = b, x = 0, x = a 7 Evaluate FN S ds where F= zi+ xj zk and S is the surface x + = 6 included in the first octant between z = 0 and z = 5 74 If F = xz i xj + k, evaluate F dv where V is the region bounded b the surfaces x = 0, = 0, = 6, z = x, z = 4 75 ompute ( ax + b + cz ) ds over the sphere x + + z = b using Gauss's divergence theorem S 76 Verif Gauss's divergence theorem to evaluate {( x z) i x j+ zk} NdS over the surface of a cube bounded b the coordinate planes x = = z = a 77 Evaluate (cos x sin x) dx + sin x cos d, b Green's theorem, where is the circle x + = 78 Verif Green's theorem in the plane for ( x 8 ) dx+ ( 4 6x) dwhere is the region bounded b = x and = x 79 Verif Stoke's theorem for F= i+ x j, where S is the circular disc x +, z = 0 80 Appl Stoke's theorem to evaluate ( dx+ zd+ xdz) where is the curve of intersection of x + + z = a and x+ z= a 6 S Q

7 Paper III - urriculum AHARYA NAGARJUNA UNIVERSITY BA / BSc DEGREE EXAMINATION, THEY MODEL PAPER (Examination at the end of third ear, for 00-0 onwards) MATHEMATIS - III PAPER III - LINEAR ALGEBRA AND VET ALULUS Time : Hours Max Marks : 00 SETION - A (6 X 6 = 6 Marks) Answer an SIX questions Each question carries 6 marks Let W and W be two subspaces of a vector space V( F) Then prove that W W is a subspace of V( F) iff W Wor W W Prove that ever finite dimensional vector space has a basis If T : V (R) V (R) is defined as T ( x, x, x) = ( x x, x + x), prove that T is a linear transformation 4 State and prove Schwarz s inequalit 5 If A = 5t i + tj t k and B = sin t i cost j + 5t k then find i) d dt ( AB ) ii) d ( A B) dt 6 Find the angle between the surfaces of the spheres x + + z = 9, x + + z + 4x 6 8z 47= 0 at the point ( 4,, ) 7 hange the order of integration in the double integral 7 0 x e dx d and hence find the value 8 Evaluate A dr where is the line joining (0, 0, 0) and (,, ), given 9(a) 0(a) A = ( + ) i+ xzj+ ( z x ) k SETION - B (4 X 6 = 64 Marks) Answer ALL questions Each question carries 6 marks Prove that the necessar and sufficient condition for a nonempt subset W of a vector space V(F) to be a subspace of V is that ab, F, αβ, W aα bβ W + (b) Show that ( 0,, ),(,, ),( 00,, ),( 0,, ) are linearl dependent If W and W are two subspaces of a finite dimensional vector space V(F) then prove that dim ( W + W ) = dim W + dim W dim ( W W ) (b) Show that the vectors α = (,,), α = ( 0,, ), α = (, 0, )form a basis of R and express (4, 5, 6) in terms of α, α, α (a) If T is a linear transformation from a vector space U(F) into a vector space V(F) and U is finite dimensional then prove that rank (T) + nullit (T) = dim U

8 Paper III - urriculum (b) Let T : V (R) V (R) be defined b T( x, x, x ) = ( x + x, x, + x, x + x + 4x ) (a) (b) (a) Prove that T is invertible and find T State and prove Bessal s inequalit Appl the Gram-Schmidt process to the vectors {(,, ), (,, ), (,, )} to obtain an orthonormal basis for V (R) with the standard inner product If A, B are two differentiable vector point functions then prove that div( A B) = B ( curl A) A ( curl B) (b) Find the directional derivative of ϕ = x + z + zx at the point (,, 0) in the direction of i+ j+ k 4(a) If F is a differentiable vector point function then prove that curl ( curl F) = grad( div F) F (b) Find div F and curl F where F = x i+ x zj z kat (,,) 5(a) State and prove Gauss s divergence theorem (b) Verif Green s theorem in the plane for ( x 8 ) dx+ ( 4 6x) d where is the region 6(a) bounded b = x and = x State and prove Stoke s theorem (b) Evaluate x ( x + ) dxd over the area between = x and = x 8 Q

9 9 Paper III - urriculum AHARYA NAGARJUNA UNIVERSITY BA / BSc DEGREE EXAMINATION, PRATIAL MODEL PAPER (Practical Examination at the end of third ear, for 00-0 onwards) MATHEMATIS - III PAPER III - LINEAR ALGEBRA AND VET ALULUS Time : Hours Max Marks : 0 Answer ALL questions Each question carries 7 marks 4 7 = 0 M (a) Let F be a field and a, a, a be three fixed elements of F Show that W = {( x, x, x): x, x, x F and ax + ax + ax = 0} is a subspace of V( F) (b) Show that the vectors (,,),( 0,, ),(,, ) form a basis for R (a) If α = ( a, a), β = ( b, b) V ( R) then show that ( αβ, ) = ab ab ab + 4 ab defines an inner product on V (R) (b) Let T: R R be a linear transformation defined b T( x, x, x) = ( x, x x, x+ x + x) Is T invertible? If so find a rule for T like the one which defined T (a) Find the angle between the surfaces x + + z = 9 and x + z = at (,, ) (b) If F = grad( x + + z xz) then find divf, curlf 4(a) Evaluate (cos x sin x) dx + sin x cos d, b Green's theorem, where is the circle x + = (b) ompute ( ax + b + cz ) ds over the sphere x + + z = b using Gauss's divergence theorem S Written exam : 0 Marks For record : 0 Marks For viva-voce : 0 Marks Total marks : 50 Marks