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- Prove that in a given vector space V, the zero vector is unique. Suppose, by way of contradiction, that there are two distinct additive identities 0 and u,. Which of the following statements are then true about the vectors 0 and u,? (Select all that apply.) O The vector 0 + u, is not equal to u, + 0. O The vector 0 + u, is equal to un: O The vector 0 + u, is not equal to 0. O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is equal to 0. O The vector o + u, is not equal to u: Which of the following is a result of the true statements that were chosen and what contradiction then occurs? O The statement u, + o 0, which contradicts that u, is an additive identity. O The statement u, +0 # 0 + u, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 U, which contradicts that O is an additive identity. O The statement u, + 0 + 0, which contradicts that u, must…3 : Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this i=1 purpose, compute the Hessian matrix of the function f(x).Program the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)
- . The determinant of an n X n matrix can be used in solving systems of linear equations, as well as for other purposes. The determinant of A can be defined in terms of minors and cofactors. The minor of element aj is the determinant of the (n – 1) X (n – 1) matrix obtained from A by crossing out the elements in row i and column j; denote this minor by Mj. The cofactor of element aj, denoted by Cj. is defined by Cy = (-1y**Mg The determinant of A is computed by multiplying all the elements in some fixed row of A by their respective cofactors and summing the results. For example, if the first row is used, then the determi- nant of A is given by Σ (α(CI) k=1 Write a program that, when given n and the entries in an n Xn array A as input, computes the deter- minant of A. Use a recursive algorithm.USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.Bu = ƒ and Cu = f might be solvable even though B and C are singular. Show that every vector f = Bu has ƒ1 + ƒ2+ ……. +fn = 0. Physical meaning: the external forces balance. Linear algebra meaning: Bu = ƒ is solvable when ƒ is perpendicular to the all – ones column vector e = (1, 1, 1, 1…) = ones (n, 1).
- Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n xn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable. Find the eigenvalues. (Enter your answers as a comma-separated list.) Is there a sufficient number to guarantee that the matrix is diagonalizable? O Yes O No Need Help? Read itLet A be an m × n matrix with m > n. (a) What is the maximum number of nonzero singular values that A can have? (b) If rank(A) = k, how many nonzero singular values does A have?A What is the product of matrices M=(1,0)(0,-3) and N=(0,1)(0,-3)? B What is the difference of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? C What is the a(3) term of the sequence {a(n)} where a(n) = 2n - 6? 1.D What is the range of a function that assigns to a 16-bit bit string the number of one bits in the bit string? E What is the sum of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? F If f(x) = 2x + 3, find f(S) if S={0,2,5}. G What is the transpose of matrix M=(1,0,2)(0,-3,1)?
- If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePythonIs W a subspace of the vector space? If not, state why. (Select all that apply.) w is the set of all vectors in R° whose components are Pythagorean triples. (Assume all components of a Pythagorean triple are positive integers.) O w is a subspace of R°. O w is not a subspace of R because it is not closed under addition. O w is not a subspace of R³ because it is not closed under scalar multiplication.4. Consider the set V of vectors (x, X2, X3, X4) ER such that X1 + x3 = 0 and x, + x4 = 0. a) Prove that V is a subspace of R*. b) Give a basis and the dimension of V.