31 Vector Calculus: Divergence Theorem Task: Refer to Question 31 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing 32 Probability and Statistics: Bayes' Theorem Task: Refer to Question 32 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC44OAZF/view?usp=sharing

7. Rank and Nullity: 1 2 1 Let A - 2 4 2 a) Find the rank of matrix A. 3 6 3 b) Find the nullity of matrix A. c) Show that the rank-nullity theorem holds for matrix A. 8. Change of Basis: Consider the vector space R² with the standard basis {e1 = (1,0), e2 = (0, 1)} and a new basis {v1 = (1, 1), V2 = (1, −1)}. a) Find the change of basis matrix from the standard basis to the new basis. b) Express the vector w = (2, 3) in the new basis {V1, V2}.

26 Numerical Methods: Newton-Raphson Method Task: Refer to Question 26 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing

- Let V = R4, and let the set S = {(1, 2, 3, 4), (0, 1, 2, 3), (0, 0, 1, 2)} be a subset of V. a) Show that S is linearly independent. b) Find the dimension of the subspace spanned by S. c) Find a basis for the orthogonal complement of the subspace spanned by S.

Let A be a 4 x 4 matrix with the following characteristic polynomial: a) Find the eigenvalues of A. P(A) = 46A3 + 11λ² – 6A b) Find the geometric multiplicity of each eigenvalue. c) Determine the diagonalizability of A.

43 Game Theory: Nash Equilibrium Task: Refer to Question 43 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing 44 Tensor Calculus: Covariant and Contravariant Tensors Task: Refer to Question 44 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing

Let A = 1 2 1 242 3 6 3 a) Find the rank of matrix A. b) Find the nullity of matrix A. c) Show that the rank-nullity theorem holds for matrix A.

9. Spectral Theorem: Let A be a symmetric matrix A- (14) a) Find the eigenvalues and eigenvectors of A. b) Show that A is diagonalizable. c) Verify the spectral theorem by diagonalizing A.

Given the system of linear equations 2xyz=1, x-3y+2z = 4, 3x+2y+3z = 5 a) Represent the system as Ax = b where A is a matrix, and solve for x using Cramer's rule. b) Compute the determinant of matrix A and discuss its implications on the solvability of the system.