BSc CS Sem 4 BSc CS Semester 4 (2018 2019) May 2019 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) Which of the following commands will create a list?
   • a) list 1 = [] c) list 1 = d) All of these
   • ii) The dot product of (1, 2,
   • iii) The dot product of (1, 2, 3) and (-1, 1, 0) is
   • iv) A linear equation with right hand side is equal to zero is called
   • v) A vector whose norm is 1 is called vector
   • a) Null b) Besis c) Unit d) none of these
   • (b) Fill in the blanks for the following questions
   • i) Two vectors are said to be orthogonal if angle between them is
   • ii) The output when we execute list(“Hello”) is
   • iii) Set of all linear combinations of vectors is called
   • iv) If all the elements of a matrix have zero value is called as
   • v) To add a new element to a list we use command
   • (c) Answer the following questions
   • i) If u=(1, 2, -1) and v = (3, 2, find norm u and norm v
   • ii) Define the term Inner Product Space
   • iii) Solve +
   • iv) Define the term Characteristic equation
   • v) Find dot product of (1, 5), (4, )
   • Q.P. Code: 34331
4. Q2 Attempt the following (Any THREE) (15M)
   • (a) Find the square root of complex number 8 — 61
   • (b) Determine whether 2, 2), v2=(0, 0, 3) and v3=(0, 1, 1) span vector space
   • (c) Write a Python program to find conjugate of a complex number
   • (d) Are the following vectors are linearly dependent
   • (e) Express in polar and exponential form 1+ i V3
   • (f) Check whether the set of all pairs of real numbers of the form (1, with operation (1, y) + (1, y’) = y + y’) (1, y) = (1, ky) is a vector space
5. Q3 Attempt the following (Any THREE) (15M)
   • (a) Find the angle between the two vectors a = (2,3,4) and b=(1,—4,3) in
   • (b) Let Compute the following if they exists
   • (c) Write a python program to enter a matrix and check if it is invertible if invertible exists then find inverse
   • (d) Check whether the set of functions are Linearly independent?
   • (e) Consider Subspace {(x, y,w,z) = 0} and =w,y = z} Find a basis and dimension of
   • i) ii) iii) N
   • (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 the following (Any THREE) 15 marks
   • (a) Solve the following system by Gaussian elimination method
   • (b) Find the orthonormal basis for subspace IR* whose generators are
   • (c) Let a = (3,0), b = (2,1) find vector in span {a} that is closet to b is and
   • Q.P. Code: 34331
   • (d) Verify Pythagorean Theorem for u = (1, 0, 2, and v = (0, 3, 4, 2)
   • (e) Find inner product, angle, orthogonality for
   • (f) Write a python program to find orthogonal projection u on v
7. Q5 Attempt the following (Any THREE) 15 marks
   • (a) Find eigen Values and eigen vectors of
   • (b) Express the following as a linear combination of vi=(—2, 1,
   • (c) Let > |R? bea linear map defined by f (x,y,z) = (x+2y-z, x+y-2z)
   • (d) Let S be a subset of vector space V. Prove that S+ is a subspace of V
   • (e) Fill the table

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