BSc Mathematics SEM II 2015 16 2015-16 ATKT MATHS 2 2015.16 Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q3 For Q.4 Attempt any three.(each mks)
   • (a) Attempt any one [Each 3]
2. Q1 Solve following equations using Gaussian elimination method
3. Q2 Write the note on elementary transformation and define with example
   • i) Transpose of matrix
   • iv) Upper and lower triangular matrix
4. Q1 (b) Attempt any three
5. Q1 Find Parametric equation of a plane passing through points (1,2,3), (4,5,6),
6. Q2 Give geometric interpretation of solution of system of m homogeneous linear equations in n unkowns
7. Q3 Define an invertible matrix and prove that = where and transpose of A respectively
8. Q4 Define addition and multiplication of matrices and find AB, BA for following
9. Q1 Prove that (R[x], is a vector space over where R[x] = Set ofall polynomials in x with real coefficients
10. Q2 Let V be a veetor space over R and W bea nonemply subset of V. Then prove that W is a subspace of V iffax + by W whenever xy W, abe R
    • (b) Attempt any three. [Each 4] Show that of finitely many subspaces space
11. Q2 Express given vector x = (1,2,0) as linear combination of given vectors
12. Q3 Pind Linear Span of = of R? 3 marks
13. Q4 Define a subspace W ofa vector space V and prove that W xyz asubset of R3 is a subspace of
14. Q3 (a) Altempt any one [Each 8}
15. Q1 Prove that rotation vector X through an angle @ in anticlockwise >
16. Q2 State Rank Theorem and verify it for following
17. Q3 (b) Attempt any three. [Each 4]
18. Q1 When is the map T:V > U_ where are vector spaces over R said to be a linear transformation? check whether following map is a linear transformation or not
19. Q2 Let V,U bea vector spaces over R. T:V — U is such that T(ax + by) = + bT(y) Vx yeV, abeR then prove that T is a linear Let be a vector spaces over R and T:V > be a linear transformation then
    • i) KerTisa subspace of
    • ii) subspace of
20. Q4 Let V,U he vector spaces over IR. > linear transformation then prove OA (a) Attempt any three (Each 5] the row form and reduce following matrix in row
21. Q3 Ifa # 0,ae then show that + a), independent subset of IR
22. Q4 Express the function 2t + as lincar combination of
23. Q5 Prove that every finitely generated vector space has a fin
24. Q6 Prove that Sum of Linear Transformation is also a Line

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