64130-mt----linear Algebra

NR

Code No: 64130/MT M.Tech. – I Semester Supplementary Examinations, September, 2008 LINEAR ALGEBRA (Communication Systems) Time: 3hours

1.a)

b)

Max. Marks:60

Answer any FIVE questions All questions carry equal marks --1 1 Let R, R be rings. Let R × R be the set of all pairs (x, x1 ) with x ∈ R and x1 ∈ R1 . Show that one can make R × R1 into a ring by defining addition and multiplication component wise. In particular what is the unit element of R and R1 ? Let V be a finite dimensional vector space over the field k. Let U,W be subspaces and assume that V is the direct sum U ⊕ W. Show that the dual space V * is equal to the direct sum U ⊥ ⊕ W ⊥ .

2.a) Let V be a vector space over R and f: V → R, g: V → R two linear mappings. Let F: V → R 2 be the mappings defined by F(v) = (f(v), g(v)). Show that F is linear. Generalize. b) Let F: V → W be a linear map whose kernel is {0} if v1. . . vn are linearly independent elements V then prove that F(v1) . . . . F(vn) are linearly independent elements of W. 3.a) b)

4.a)

b)

Slove 10x – 7y + 3z + 5u = 6, -6x + 8y – z – 4u =5 3x + y + 4z + 11u = 2, 5x – 9y – 2z + 4u = 7 by Gauss – Jordan elimination method. Reduce the following matrix to reduced echelon form 1 3 5 7  A =  2 4 6 8   3 5 7 9  Apply elementary row operations to transform the following matrix first into echelon form and then into reduced echelon form  0 3 −6 6 4 −5  3 −7 8 −5 8 9     3 −9 12 −9 6 15  Find a matrix  similar to the matrix Contd…2.,

Code No: 64130/MT

::2::

 −1 2 −2  A=  1 2 1   −1 −1 0 

5.a)

Let V be a vector space over K and let { v1, v2, …vn } be a basis of V. Let W be a vector space and let {ω1 , ω2, ....ωm } be a basis of W. Prove that the mapping LA :V → W is a linear mapping.

b) Let F: V → V be a linear map and let B and B1 be bases of V then there exists an invertible matrix N such that 1 M BB1 ( F ) = N −1M BB ( F ) N 6.a)

b)

Let A1 ,. . . . , Am be square matrices, of sizes d1 × d1 ,....., d m × d m respectively. Show that the determinant of the matrix  A1 0.......0  A=  0 A2 ......0   0 0....... Am  Consisting of blocks A1 ,. . . . , Am on the diagonal is equal to the product D(A) =D( A1 ) . . . . D( Am ) Let V be the space 2 × 2 matrices over the field K and let B ∈ V. Let LB : V → V be the map such that LB (A) = BA, show that LB is linear and that Det( LB ) = ( DetB )

2

Let V be a vector space over K and let A: V → V be an operator. Let v1 , v2. . . . vm be eigen vectors of A with eigen values λ1 , λ2 ,....., λm respectively. Assuming that these eigen values are distinct, prove that v1, v2. . . . vm are linearly independent. b) Let V be an n dimensional vector space over the complex numbers and assume that the characteristic polynomial of a linear map A: V → V has n distinct roots. Show that V has a basis consisting of eigen vectors of A.

7.a)

8.a) Find the characteristic 3 2 the matrix A=  0 1  0 1 b)

polynomial, eigen values and eigen vectors of 1 2  −1

State and prove Cayley-Hamilton Theorem. &_&_&_&