Let A e Rmxn. The linear system Ax = b can have either: (i) a unique...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
2 x [b] Consider the following linear system of equations AX =B : (i) Determine a basis for the row space of A. (ii) Compute the Rank of the augmented matrix (A:B), then use it to classify the solution of this system (Unique - Many -No: solution). (iii) Is the matrix A diagonalizable? Explain your answer and verify the similarity transformation.
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
Write each statement as True or False (a) If an (nx n) matrix A is not invertible then the linear system Ax-O hns infinitely many b) If the number of equations in a linear system exceeds the number of unknowns then the system 10p solutions must be inconsistent ) If each equation in a consistent system is multiplied through by a constant c then all solutions to the new system can be obtained by multiplying the solutions to the original...
3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 7z = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working. 2. Let v= [6, 1, 2], w = [5,0, 3), and P= (9, -7,31). (i) Find a vector u orthogonal to both v and w....
10. Determine the values of k for which the system of linear equations has (i) no solution vector, (ii) a unique solution vector, (iii) more than one solution vector (x, y, z): (a) kx+ y+ z= (b) 2x + (k-1)y + (3-k)2-1 2y + (k-3): = 2 x+ky + z = 1 -2y+ x 2x + ky- z =-2 (c) x + 2y + k= 1 (d) -3z =-3 10. Determine the values of k for which the system of...
Have som problem solving this problems in MATHEMATICAL METHODS 2. Can someone help me solve and explain this problems? Problem 1. Solve the following systems of linear equations: 4x+3y + z = 9 3x+y-z = 3 x4 Problem 2. Find the values of the parameters a and p for which the system x+5y = 6, has: (i) a unique solution, (ii) infinitely many solutions, (iii) no solutions. Hint: Use the following fact: a system with as many unknowns as equations...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
1. Verify that the following linear system does not have an infinite number of solutions for all constants b. 1 +39 - 13 = 1 2x + 2x2 = b 1 + bxg+bary = 1 2. Consider the matrices -=(: -1, -13). C-69--1--| 2 -1 0] 3 and F-10 1 1 [2 03 (a) Show that A, B, C, D and F are invertible matrices. (b) Solve the following equations for the unknown matrix X. (i) AXT = BC (ii)...