BSc CS Sem 4 ATKT 2018-19 ATKT Linear Algebra Using Python Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2) Figures to the right indicate marks

1. Q3 Illustrations, in-depth answers and diagrams will be appreciated
2. Q4 Mixing of sub-questions is not allowed
3. Q1 Attempt All(Each of 5Marks) (15M)
   • (a) Multiple Choice Questions
   • i) The set of all linear combinations of vectors V1, V2,., Vnis called the of the vectors
   • ii) Nullity of T is the dimension of of T
   • a) Image c) Rank d) none of the above
   • iv) In GF (2), 1+1+0+1 =
   • v) For any homogenous system is a trivial solution
   • a) Zero b) c) one d) none of the above
   • (b) Fill in the blanks
   • i) A vector whose norm is one is called vector
   • ii) A vector space together with inner product is called space If most of the element of a matrix have zero value is called
   • iv) The absolute value of 3+6i =
   • v) Inverse of a matrix is
   • (c) Define
   • i) Dot product
   • v) Dimension
   • Q.P. Code: 34333
4. Q2 Attempt the following (Any THREE) (15M)
   • (a) Find the Square root of 21 - 20i, where i= V—1
   • (b) Consider the following system of equation and find the nature of solution without solving it
   • (c) Solve the following system by backward substitution method
   • (d) Let W, and are two subspaces of V then prove that is also a subspace of V where V is a vector space on IR
   • (e) Write a python Program for rotating a complex number
   • (f) Which of the following is.a set of generators of
   • ii) {( (0,0,1)}
5. Q3 Attempt the following (Any THREE) (15M) Find the null space of matrix 6
   • (b) Let f: U> V is a linear transformation then show that kerf = {0} iff f is injective
   • (c) Find the co-ordinate representation of vector v = in terms of the vectors [1,1,0, 1], [0,1,0, 1] and [1,1,0, 0] in GF (2)
   • (d) Find the angle between the two vectors a = (2,3,4) and b=(1,—4,3) in
   • (e) Consider Subspace U, {(x, y,w, = 0} and = z} Find a basis and dimension of
   • i) iii) A
   • (f) If V and W are two subsets of a vector space V such that U is a subset of W then show that is a subset of where are annihilator of
6. Q4 Attempt the following (Any THREE) 15 marks
   • (a) Let u and v are orthogonal vectors then prove that for scalars a,b
   • (b) Explain Internet Worm
   • (c) Write a program in python to final gcd (240,24)
   • (d) Solve the following system by Gaussian elimination method
   • (e) Find the orthonormal basis for subspace IR* whose generators are
   • (f) Let a = (3,0), b = (2,1) find vector in span {a} that is closet to b is and
7. Q5 Attempt the following (Any THREE) 15 marks
   • (a) Let T: > |R? bea linear map defined by f (x,y,z) = (x+2y-z, xt+y-2z)
   • (b) Fill the table Find eigen values and eigen vectors of 3
   • (d) Let S be a subset of vector space V. Prove that S+ is a subspace of V
   • (e) Check whether the following set {(1,1,0),(0,1,1),(1,1,1)} is linearly Independent or not

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