BSc Sem IV 2014 2015 2015 Math Ll Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q3 For Q.4 Attempt any three.(each 5 mks)
2. Q1 (a) Attempt any one {Each 8]
3. Q1 Verify Rank-Nulity Theorem for T: > R? defined by T(x, y,z) = (x + y,2)
4. Q2 Let V,U be vector spaces over R. If U isa linear transformation then prove Also Define a Linear Transformation and Check whether following is a linear transformation or not
5. Q1 (b) Attempt any three. [Each 4]
6. Q1 Let V, U be a vector spaces over R and T: U > V bea linear transformation then define Image of T (Img T) and prove that Img T is a subspace of V
7. Q2 Define a matrix associated with linear transformation and find matrix of linear transformation for following linear transformation T: IR? > y) = (x + y, 2y, x — y) with respect to natural basis
8. Q3 Let V, be m-dimensional and n-dimensional vector spaces over and B = = be ordered bases of V,V' respectively Prove that if > V’Tz:V — V’ are linear transformation Then
9. Q4 Define row rank, column rank and rank of a matrix A and find rank of
10. Q2 (a) Attempt any one [Each 8]
11. Q1 Define the determinant of a matrix of order n and derive the fo
12. Q2 Prove that if A is matrix of order n then det A = det where A? = transpose of
13. Q2 (b) Attempt any three. [Each 4]
14. Q1 Define Determinant of a matrix using Laplace Expansion and use Laplace Expansion by row to find determinat of following matrix
15. Q2 Define Vandermonde determinant and solve
16. Q3 Define linearly independant and linearly dependant vectors in R” Check whether following vectors are linearly independant or linearly dependant
17. Q4 Find inverse of a matrix using adjoint method
18. Q3 (a) Attempt any one [Each 8] ) Find eigen values and eigen vectors
19. Q2 Verify Cayley Hamilton Theorem and hence find if exist,
20. Q3 (b) Attempt any three, [Each 4]
21. Q1 Prove that Zero js eigen value of a matrix iff matrix is singular,
22. Q2 Define an eigen value of a matrix and prove that the eigen values of a matrix and its transpose are same
23. Q3 Define similar matrix and prove that if A & P aren n matrices and P is nonsingular then A &P-1AP have same eigen values
24. Q4 Prove that the eigen values of a diagonal matrix are diagonal elements
25. Q4 (a) Attempt any three [Each 5]
26. Q1 Let V be a finite dimensional vector space over Rand T:V > V bea linear transformation Then prove that T is invertible iff T is one-one,
27. Q2 Let U,V, W be vector spaces over R and + V,S:V + W bea linear transformation Then prove that the composition map SoT:U W isalsoa linear
28. Q3 Find volume of parallelepiped bounded by three vectors Vj, V2,V3
29. Q4 Find area of parallelogram formed by edges V,, where
30. Q5 Prove that eigen vectors corresponding to distinct eigen values are linearly
31. Q6 Define the following term
    • i) Eigen value of a matrix ii) Eigen vector of a matrix Characteristic polynomial of a matrix

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