UNIT-II-Vector spaces and subspaces, L.I, L.D. Basis and dimension. 1. Define a vector space and sub space.P.T the necessary and sufficient condition for a non empty subset W of a vector space V(F to be a subspace of V is for all a,b Є F and α,β Є W==> aα+bβ Є W . 2. Show that the intersection of two subspaces of a vector Space is also a subspace. 3. If W 1,W 2 are subspaces for V(F) then show that W1UW 2 is a subspace if and only if W 1 is subset of W 2 or W2 is subset of W 1. 4. If V(F) is a vector space and S V then prove that L(S) is subspace of V. 5. Define linear independence and linear dependence of Vectors in a vector space V. Prove that: a) superset of L. D set is L.D. b) subset of L.I set is L.I.. 6. Prove that two vectors are linearly dependent if one of them is a scalar multiple of the other. 7. Prove that a system containing zero vector is L.D. 8. Prove that a system containing single zero vector is L.D. 9. P.T. the set of non-zero vectors α1, α2, α3,…….. αn, of V(F) is linearly dependent iff some αk 2≤k≤n is a linear combination of the preceding vectors. 10. Define basis of a vector space. P.T. there exists basis for each finite dimensional vector Space. 11. State and prove Dimension theorem. (OR) P.T. any two bases of a finite dimensional vector space V(F) have same number of elements. 12. P.T. Every L.I. subset of a finitely generated vector space V(F) is either a basis of V or it can be extended to form a basis of V. 13. Define basis of a vector space. If dim(V)=n then show that every set of n vectors in V is a basis if it is linearly independent. 14. Define dimension of a vector space. If V is a finite dimensional space and U,W being subspaces of V prove that: dim (U+W)=dim U+ dim W- dim(U W). 15. All the problems based on: Sub spaces, Dimension of sub spaces ,Linear span, L.I, L.D, . II- Linear transformation, Rank and nullity. 1. a) Define the rank of T and nullity of L.T.; b)* State and prove rank and nullity theorem. 2. *Let U and V be vector spaces. Let T: U→ V be L.T. Then prove that: a) N(T) is a sub space of U. b) R(T) is a subspace of V. 3. Let V and W be vector spaces. Let T: V→ W be L.T. Then show that T is one-one if and only if N(T)=0. 4. *Let V and W be vector spaces of equal (finite) dimension and let T: V→ W be linear. Show that the following are equivalent. 1) T is one-one 2) T is onto 3) rank(T)=dim(V) 5. S.T. the operator Ton R defined by T(x,y,z) = (x+z, x-z, y) is invertible and find similar rule defining T-1 3 6. Let T be a linear operator on V3(R ) defined by: T(x,y,z)=(3x+z, -2x+y, -x+2y+z . P.T. T is invertible and hence find T-1 7. Let T1 and T2 be two linear operators defined on V3(R ) byT1 (a,b,c)=(a+b, 2b, 2b-a ), T2 (a,b,c)=(3a, a-b, 2a+b+c )for all a,b,c € V3(R ) show that T1 T2≠ T2 T1 8. Describe explicitly the linear transformation T: R2 R2 such that T(2,3)=(4,5) and T(1,0)=(0,0). 9. Define a linear transformation. S.T. T: V3(R ) V2(R ) defined as T(a,b,c) = (3a-2b+c, a-3b-2c) is a linear transformation. 10. Find the kernel of the L.T. T: R2 R2 defined as T(1,0)= (1,1) and T(0,1)= (-1,2). 11. Find a linear transformation whose range is spanned by (1,2,0,-4) , (2,0, 1, -3). 12. If T: V4(R ) V3(R ) is a L.T. defined by T(a,b,c,d) = (a-b+c+d, a+2c-d, a+b+3c-3d) for a,b,c,d in R , then verify rank nullity theorem. UNIT 2 I- Eigen values and Eigen vectors. 1) Show that the characteristic vectors corresponding to distinct roots of a matrix are linearly independent. 2) Define diagonalizable matrix. State and prove necessary and sufficient condition for an nXn matrix to be diagonalizable. 3) State and prove Cayley Hamilton theorem. 4) All the numerical problems on the following model: 1) Define the characteristic polynomial of a matrix A. Find the characteristic equation , values and vectors of the matrix 0 1 1 0 2 3 iii) 2 4 i) ii) iv) 0 0 0 i 3 13 3 1 0 2 3 3 2 4 2 0 1 v) 1 1 1 vi) 2 0 2 viii) 4 1 4 2 5 4 2 3 2 0 1 2 9 4 4 2) Test for diagonalizability of: i) 1 2 over C ii) 8 3 4 over R 0 1 16 8 7 3) Verify Caley Hamilton theorem for the following matrix A and hence find A-1 2 1 1 2 2 1 1 3 1 1 2 1 1 2 1 1 2 2 II- INNER PRODUCT SPACES 1) State and prove Cauchy –schwarz inequality. 2) Verify Cauchy –schwarz inequality for the vectors x=(2,1+i,i) ,y=(2-i,2,2i+1) in c3. 3) State and prove Triangle inequality. 4) State and prove Bessel’s inequality. 22 2 2 2 2 5) If α ,β are vectors in an IPS V then prove that 6) Define the norm of a vector space in an inner product space V. f x and y are vectors V then prove that 1 1 i) x y x y 2 x 2 y x y x y 2 2 2 2 2 2 ii) Re(x,y)= 4 4 7) Define an Inner product on vector space V. If 𝛼 = (𝑎1 , 𝑎2 ), 𝛽 = (𝑏1 , 𝑏2 ) ∈ 𝑉2 (𝑅). Define : (𝛼, 𝛽) = 𝑎1 𝑏1 − 𝑎2 𝑏1 − 𝑎1 𝑏2 +4𝑎2 𝑏2 . S.T. all the postulates of an inner product hold good. 8) Suppose that α, β are vectors in an IPS V, P.T. iff ( , ) α and β are linearly dependent 9) If in an inner product space then prove that the vectors α, β are linearly dependent. Give an example to show that the converse of this statement is false. 10) P.T. if α, β are vectors in an unitary space then (i) 4(𝛼, 𝛽) = ‖𝛼 + 𝛽‖2 − ‖𝛼 − 𝛽‖2 + 𝑖‖𝛼 + 𝑖𝛽‖2 − 𝑖‖𝛼 − 𝑖𝛽‖2 . (ii) (𝛼, 𝛽) = 𝑅𝑒(𝛼, 𝛽) + 𝑖𝑅𝑒(𝛼, 𝑖𝛽) 11) Define orthogonal and Orthonormal subsets of an IPS. S.T. { (1,1,0),(1,-1,1),(-1,1,2)} is orthogonal. Is it orthonormal? 12) Let S= { 1 , 2 , 3 ,........ n } be an orthogonal set of non zero vectors in an Inner Product space V. If , k k n a vector β in V is in the span of S then β= k 1 k 2 13) Let S= { 1 , 2 , 3 ,........ n } be an orthonormal set of non zero vectors in an Inner Product space V. , k k n If a vector β in V is in the span of S then β= k 1 14) Let V be an IPS and let S be an orthogonal subset of V consisting of nonzero vectors. S.T. S is linearly independent. 15) Let V be an inner product space and let S be an orthonormal subset of V. Show that S is linearly independent. 16) If V is a finite dimensional inner product space and { 1 , 2 , 3 ,........ n } is an orthonormal basis for V. S.T. for any vectors α ,β in V, (α, β)= n i 1 ( , i )( , i ) . 2 2 2 17) Prove that two vectors α,β in a real inner product space are orthogonal iff . Is it true for complex inner product space? Justify. 18) If x, y are two vectors in V(F) then S.T.x,y are orthogonal if and only if ax by a x b y fora, b F . 2 2 2 2 2 19) Find a unit vector orthogonall to (4,2,3) in R3. 20) Find two mutually orthogonal vectors each of which is orthogonal to the vector: α= (5,2,-1) of V3(R ) with respect to the standard inner product. 21) Find two L.I. vectors each of which is orthogonal to (1,1,2). 22) Verify that the vectors: ( 1 , 2 , 2 ), (( 2 , 1 , 2 )( 2 , 2 , 1) form an orthonormal basis for V3( R ) 3 3 3 3 3 3 3 3 3 w.r.t the standard inner product. 23) *Describe Gram schmidt’s orthogonalisation process. Apply it to obtain an orthonormal basis : 1) of V3( R ) 2) (2,0,1),(3,-1,5) and (0,4,2) for 3) (1,0,0) ,(1,1,0), (1,1,1) of R3. 4) 1,x, x2 of P2(R ) 5) {(1,0,1),(0,1,1),(1,3,3)} of V3( R ) 24) Using Gram Schmidt process compute orthogonal vectors corresponding to the L.I.vectors {(1,0,1,0),(1,1,1,1),(0,1,2,1)}. 25) Define orthogonal compliment of a sub space W of an inner product space V(F) and prove that it forms a subspace. UNIT 3(MULTIPLE INTEGRALS) All the problems given in the text book needed to be practiced UNIT 4 (VECTOR CALCULUS) 1) P.T: ∇. (A X B) = B. (∇ X A)-A. (∇ X B) OR Div (A X B)=B. Curl A-A. Curl B 2) P.T: ∇𝑋 (A X B) = (B.∇)A- B(∇. A)- (A.∇) B+ A (∇.B). (OR) curl (A XB) = A div B - B div A + (B. ∇ ) A - (A. ∇) B 3) Prove that: grad (A.B) = (B.∇)A+ (A.∇)B+B X (∇XA) + A X (∇XB). 4) ∇(∇φ) = ∇2 φ 5) ∇𝑋∇φ = 0 i.e Curl of grad φ is zero vector. 6) Prove that: ∇ X (∇ X A) = ∇ (∇ .A) - ∇2 A 7)* State and prove Green's Theorem. 8) *State and prove Stokes Theorem. 9) *State and prove Gauss Divergence Theorem. 10) All the problems based on: 1) Vector differentiation, Vector differential operators, Vector Integration, 2) Line Integrals, Surface integrals, Volume Integrals, 3) Verification of Green’s, Gauss and Stokes theorem.