linear algebra 2005-06(3 q.p.)

8/3/2019 Linear Algebra 2005-06(3 Q.P.)

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Page No... 1 LDE 11I NEW SCHEMEUSN

7

First Semester M.Tech Degree Examination , January/February 2005Digital ElectronicsLinear Algebra

Time: 3 hrs.]Note: 1. Answer any FIVE full questions.

2. All questions carry equal marks.

JMax.Marks : 100

1. (a) If V is the vector space over the field F. Prove that the intersection of any familyof subspaces of Vis a subspace of V (8Mrks)(b) Prove that the subspace spanned by non empty subset S of a vector space Vis the set of all linear combinations of vectors inS (6Mrks)(c) In the vector space R3(R), let a = (1, 2, 1), (.3 = (3,1, 5) -y = (3,-4,7). Provethat the subspace spanned by S = {a, 0} and T = {a,,3, y} are the same.

( 6 Ma rk s )2. (a) If Wl and W., are finite dimensional subspaces of a vector space V, then show

that Wt + W2 is finite dimensional and dim TV1 + dim W, = dim (W n W,) +dim (Wl +W Z) (a Marks)

j7( F1 he finite riim --inr1al vertrnr --Hart-anri let In h rv-1 he an orri


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Page No... 2 LDE112100

(b) Find the minimum polynomial m(t) of A = 0 2 . 0 0 (8 Marks)0 0 -2 4(c) Let A = ( 2 3 ) rind all eigenvalues of A and corresponding eigenvectors and

show that P-1AP is diagonal . (6 Marks)6. (a) Let W1, W2,... W, are subspaces of V and for { wz1, will ...W ini} is a basis ofW i f o r i = 1, 2, ...r then V is the direct sum of Wi iff the union

B = {w11, ...wlnl, ...wr1, ....wrnr} is a basis of V. (a Marks)(b) Let T : V -p V be linear, suppose for V E V, Tk(v) = 0 but Tk-1(v) f- 0 prove

i) the set s = {v,T(v), Tk-1(v)} is linearly independentii) The subspace W generated by S is T - invariant.iii) The restriction t of T to W is nil potent of indexk (6Mrks)

(c) Determine all possible Jordan cannonical forms for a linear operator T : V -. Vwhose characterestic polynomial is At = (t - 2)3 (t - 5)2. (6Mrks)

7. (a) Prove that every inner product vector space is a normed vector space . (a marks)(b) An orthonormal set {u1, u,, ...Ur} is linearly independent and for any v E V,

show that, the vector.


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a a1-., kr) be a vector space and let )31 i Q,, ....13,E be any vectorsin W. Let T : V - W defined by T(ai) = ;3i i = 1, 2,...n. Prove that T islinear transformaion fromVtoWand is unique. ( Marks)

(c) Prove that the set S = {(1, 2, 1, (2, 1, 0), (1, -1, 2)} forms a basis for V 3 (F).(6 Marks)

3. (a) Define rank and nullity of linear transformation T : V(F) -. W(F). and Vis finite dimensional. Prove thatRankn+Nulity(T =dmV(Marks)(b) Let V and W be vector spaces over the field F and let T : V -i W be linear

transformation. If T is invertible then the inverse function T-1 : W - V Isanearansfomon (Marks)(c) If T : R3 R3 is a linear transformation defined by T(x, y, z) = (x - 3y -

2z, x - 4z, z) is invertible find T-1 6Mrk4. (a) Sate and prove Cayley-Hamlton theorem (Marks)

(b) Find all eigenualues and a basis of each eigen space of the operator T : R3 -- R3defined by T(x, y, z) = (2x + y, y - z, 2y+ 4z) (6 Marks)

5. (a) Show that a linear operator T : V - V has a diagonal matrix representationiff its minimal polynomial m(t) is a product of distinct linear polynomials.(8 Marks!

ra.


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- , U) u 1 , , ...... , . -..1(c) Let the basis of Euclidean space R3 is {v1 = (1,1,1), v2 = (0,1,1),v3 = (0, 0, 1)} use Gram-Schmidt orthogonalization process to transform {vi}inoanorthonorml basis {u} (Marks)

1118. (a) Determine the inverse of the matrix 4 3 -1 by using partition method353and solve the following system of equations.x1+x2+x3=14x1+3x2-x3=63x1 + 5x2 + 3x3 = 4 (8 Marks)

(b) Solve the system of equations using Gauss - Seidel iteration method so as toget the solution accurate upto three decimal places.

10x1 + 2x2 + x3 = 92x1 + 20x2 - 2x3 = -44-2x1 + 3x2 + 10x3 =226Mrk

(c) Find all the eigen values and eigen vectors of the matrix using Jacobi method.12A=36M k2v1 ** * **


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Page No... I 05EC046USN FT

NEW SCHEMEFirst Semester M.Tech. Degree Examination, Dec.06/Jan. 07

Linear AlgebraTm 3hs][Mx Mrks 100Note : Answer any FIVEquestions.

1 a. If A and B are m x n matrices, show that B is row equivalent to A if and only ifB = PA, where P in the product of m x m elementary matrices. (06 Marks )

b. If A = A, A2 ... Ak where each Ai is an n x n square matrix, show that A is invertibleif and only if each A; is invertible 06Mrk

H 1 2 1 01c. If A= -103 5

1-211, find the row equivalent echelon matrix R to A and an

invertibe matrixP Such that R=PA (08Mrks)2 a. Let P be an n x n invertible matrix. Let V be an n - dimensional vector space and B an

ordered basis for V. Show that there is a unique ordered basis B of V such that[a]B=P[a]B. and[a ]B' = P'' [a ]B.(6 M a r k s )b. If W, and W2 are finite dimensional subspaces of V, then show that W, + W2 is finite

dimensional and dim W, + dim W2 = dim (W, + W2) + dim (W, n W2).( 0 6 M a r k s )

c. If (-1, 0, 0), (4, 2, 0), (5, -3, 8) are basis vectors in B', express the vector (x,, x2, x3)with respect to the basis B'. Hence express (1, 2, 3) with respect to B'. (08 Marks)

3 a. Let V be an n - dimensional vector space and let W be an n - dimensional vectorspace over F. Show that the space L (V, W) of linear transformations has dimensionm 06Mrk

b. Let V and W be n - dimensional vector spaces over F. Let B and B' be ordered basesfor V and W respectively. For each linear transformation T : V --+ W, show that thereis anm x n matrixAsuch that [ Ta ] B =a [a ]B. (6 M a r k s )

c. If a, = (1, 2 ), a2 = (3, 4) are bases for R2 and Ta, _ (3,2,1), Tae = (6,5,4). Represent Tin the matrix for m where T is a linear transformation from R2 -> R3. (08 Marks)

4 a. Let V be a finite dimensional vector space. Let B = (a,, a2, ....., a) be a basis of V.Show that there is a unique dual basis B* = { f,, f2, ..., f } for VO such that f, (aj) = Bij.Show also that for each linear functional f on V.

f= n f(ai)a.i and for each vector a in V a= fi(aei (06Marks)=1=1

b. Define the transpose T of a linear transformation T with the usual notation, if A is thematrix of linear transformation of T from V to W relative to the bases B and B1, andBM is the matrix of the transpose T of T relative to B. , B the dual bases, show thatBUM =Aji . Where BM=BM06MrksJJCond.. 2


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Page No ... 2

c. Find the dual basis of B = ((1, -1, 3),(0,1,-1),(0,3,-2) }.

1 3 51 1 01 1 21 3 3

5 a. Prove the Cayley - Hamilton theorem : A square matrix of order n satisfies its owncharacerisicspoynoma. (6 M a r k s )b. Show that an n x n matrix is diagonalizable if and only if A has n linearly independenteigen vectors 06Mrk)1 3 3

c. Diagonalize the matrix A = -3 -5 -3 using D = P"1 AP and finding P.3 3 16 a. Explain Gram - Schmidt process of orthogonalisation. 3-2 4

b. Orthogonally diagonalize the matrix a given by A = -2 6 24 2 3c. Find the least square solution of AX = b.

Where A =

7

and b =357-3

05EC046

(08 Marks)

(08 Marks)

( 0 6 M a r k s )

(07 Marks)

(07 Marks)

a. Convert the quadratic form Q(x)=x? -8x1x2 -5x2 into a quadratic form with no crossproduct term. Verify that both will have the same value at a point (xi, x2). (06 Marks)

b. Find the maximum and minimum values of Q(x) = 9x +4x2 +3x3 . Subject to theconstran XTX=I (07Mrk)59, findQRfactorizationof A (07 Marks). If A = 13 7

1 5

8 Explain the following and illustrate the same with an example.a. Characteristics polynomialb. Annihilating polynomialc. Minimal polynomiald. Principal decomposition ( Primary decomposition ). (20 Marks)


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Page No ...1USNf

NEW SCHEMEM.Tech Degree Examination, May / June 2006Linear Algebra

05EC046-] T- I

Time: 3 hrs.] [Max. Marks:100Note : 1. Answer any FIVE J411 questions.

21 a. Determine Scalar K such that (KA)T(KA) = 1, where A = 1 . Is the value unique?

-1(05 Marks)

b. Solve for the sy - 4z = 8;

ystem of equ2x

ation- 3y

s,+ 4z =1 ; 5x -8y + 7z = 1 . (05 M arks)

c. For what values of scalar K will the vector Y be in the span { V , , V 2 , V,) where1

,V2=5-3-4 ,V3= 1 and Y=-70 F - 4 -3 (10 Marks)K

3 -6 32 a. Find LU factorization of matrix, A = 6 -7 2 (10 Marks)

-17 01 l

b. Diagonalize the matrix B = L i 0 ( 1 0 M a r k s )3 a. Let T be a linear Transformation from R5 to R4 be defined asT(x, x2x3x4x5 ) _ (x, - x 3 +3x4 -x5,x2 +x4 - x,,2x2 -x3 +5x4 - x5,-x3 +x4)

Find a basis and dimension by Im TFind a basis and dimension of KerT 10Mrk

Xb. Let T y = z + y be a transformation from IR3-- R2. If V = [V,, V 2, V3] and1Y-Z1 0 0

W=[W W,]where V= 0 ,V2= 1 ,and V3= 0 and W,= U ,W2= 0 .00Find the matrix representation [ T ] with respect to V and W . Suppose we now changeVand Was V,=

I01

0V2= and V3 =

What is the new representation? Contd.... 2


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Page No... 2 05EC0460 0 -2

4 a. Given the symmetric matrix, A = 0 -2 0 diagonalize this. ( 10 M arks )-2 031 -1 -10b. Find QR factorization of matrix, A= 1 010(10Mrk)01 -1

5 a. Let T : P2 -+P2 be a linear transformation as defined by T(at2+bt+c) = (a+2b)t + (b+c)i) Is -42+2-2 in Ker T? ii) Is t2+2+1 in Range T?iii) Is Tone toone? iv) Is Tonto?v) Find basis and dimension of Im T. vi) Find basis and dimension of Ker T.vii) Varify Rank Nullity Theorem 15Mrk

b. Let W be the subspace of IR 4 with basis W 1 and W 2 where Wl =(1 1 0 1) andW2 = (0 -1 1 1). Find a basis forW 05Mrk6 a. LeU and W be the following subspaces by IR4 : U ={(a, b, c, d) : b + c + d = OlandW ={(a, b, c, d) : a + b =0, c =2d). Find the basis and dimension of) U)W iii) U f1 W iv) UUW (10 Marks)

2100b. let A =0 2 0 0 Find characteristic polynomial, minimal polynomial of this.0 0 2 0

0 0 0 5(10 Marks)7 a. Determine all possible Jordan Canonical forms of a matrix of order 6 whose minimalpoynomal is m2) =(A- 2)2. (0 M a r k s )

2-4b. Determine the Invariant subspaces of A = 5 -21 viewed as linear operator on)IR)C (10Mks)8 a. By using orthogonal projection determine the least square solution to this system of1-6-11 -2 2equation Ax =b when A = I I , b=I . (05 M arks )176

b. Verify the above result by computing least square inverse of A and using it. (05 Marks)c. Find singular value decomposition of matrix,

A= (10 Marks)

linear algebra 2005-06(3 q.p.)

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