BSc CS Sem 3 2023 2024 2024 LINEAR ALGEBRA Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks

1. Q1 Attempt the following (Any Four) (20M)
   • a) Define Modulus and argument of complex number z. calculate modulus and argument
   • b) Find square root of complex number 7+24i
   • c) Define addition, subtraction, multiplication and division of two complex number
   • d) Define linear combination of vectors in vector space V. Express (2,4,9) as a linear combination of (1,1,0), (0,2.1) and (0,1,2)
   • e) Prove all five addition properties of vector space for the set
   • f) Solve the non-homogenous linear equations
2. Q2 Attempt the following (Any F our) (20M)
   • a) Reduce the following matrix in a Row-Echelon form
   • b) Verify the subset S = {(1,2), (2,1), (1,1)} of R?is linearly Dependent
   • c) Using Gauss elimination method solve the following equation
   • d) Let T: R* > R defined by = x — y. Show that T is a linear Transformation that Every Superset of Linearly Dependent set is Linearly Dependent
   • f) Define Basis of a vector space. Show that {(1,0),(1,1)} is basis of
3. Q3 Attempt the following (Any Four) (20M)
   • a) Find minimal polynomial m(t) of —15
   • b) Use the Gram-Schmidt orthonormalization process to construct an orthonormal set of vectors from the linearly independent set { (3,1),(4,2)}
   • c) Prove that |< u,v >| u Il Il v for any two vectors in vector
   • d) Verify Cayley Hamilton Theorem for matrix Let X and be two vectors then that < + XnYn iS an inner product in
   • f) Find Bigen values and Eigen \ vectors for the matrix A= Pe: Attempt the follow
   • a) Find distance and an Ween two vectors x= (1,-2) and y=(-2,1)
   • b) Consider the basis B - (3,2)} and B’ = {(1,0), (0,1)}. If u is a vector such Define Dimension 0 with suitable example,
   • d) Explain Galoi’s f and multiplication operation 7 Express the number. ‘in the form of
   • f) Verify the set S={(x, a vector Subspace in IR with respect

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