Mathematics for Computing — DU B.Sc. (Hons) Computer Science Previous Year Question Papers

3. Parts of a question must be answered together. 1(a) Find the Rank of the following matrix using row echelon form (5) b) Find the dot product, cross product, and angle between the vectors (5) (C) Determine following transformation Tis lincar or not? (5) (d) Definc Basis. Show that the vectors (2,1,4), (1, -1,2), (3,1, -2) form a basis (5) (c) Given following matrix Use Cayley-Hamilton theorem to evaluate: (A-I) (1-6I). (1) Define Subspacc. Let Y be the set of vectors in R* of the form [a,0,b,0). (5) Prove that Y is a subspace of R*. What is Positive definite matrix? Is the following matrix positive (7) (b) Solve the following system of linear equations using Gauss Jordan (8) Calculate the curl and divergence for the following vector field at (7) Diagonalize the following matrix (8) 4 (a) For what value of 2 and u, do the following system of equations be (7) (b) Apply Gram Schmidt orthonormalization process to obtain an (8) orthonormal basis for the subspace of R3 gencrated by the vectors What is directional derivates? Find the directional derivative of (7) f(x, y,z) = x2z+×z3 - xy at the point P(1,1,- 1) in the direction of (b) Define inner product space. Consider a real vector space R2, which is (8) Show that it is an inner product space. 6 (a) Find inverse of the following matrix using Gauss Jordan Elimination (7) (b) The weather in a city change according to a Markov Chain with three (8) states: Sunny (S), Cloudy (C), Rainy (R). The Transition Matrix is: (i) If today is Sunny, find the probability that it will be Rainy after one day. (i) If today is Cloudy, find the probability that it will be Sunny afler (iii) If the initial state vector is v (0) = (0.5, 0.3,0.2), Find the state distribution after one day and after two days. 7 (a) Find the bases of row space and null space of the following matrix (7) (b) The set R? defined with addition opcration and scalar multiplication on (8) Show that R2 is a vector space over addition and scalar multiplication mare tanner gerar