2, (15 pts) For an m × n real matrix A and vectors x and b, find ▽zllAx-bll3 and derive the normal equations. 2, (15 pts) For an m × n real matrix A and vectors x and b, find ▽zllAx-bll3 and...
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
3. (15 pts) Let A be an m x n matrix with rank r, and let V = C(A). (a) V CIRP for what p? (b) What is V. in terms of a fundamental subspace for A? (c) How many vectors are in a basis for V, and how many in a basis for v 1? (d) For what m, n, and r docs Ax=b have a solution for every b? (e) Is a set of r vectors in V...
5. (10 pts) Consider the two-mass sy stem of Fig. 1. The system is free to move in x1 plane. a) Derive the equations of motion. b) Identify the mass matrix and the stiffness matrix if the displacement vector is x=1 x, x2 x3 x4 3k 4k 4k
5. (10 pts) Consider the two-mass sy stem of Fig. 1. The system is free to move in x1 plane. a) Derive the equations of motion. b) Identify the mass matrix and...
1. Derive the Lorentz transformation for the "3-vectors" É and B from the normal orentz transtormation for the ran (Helps to get it set up like a normal matrix multiplication.) Give the transforma tion for the components of the 3-vectors parallel to the boost and perpendicular to the boost. Hint: Do the calculation of F' by multiplying the matrices. Then look at compo- nents of the fields parallel and perpendicular to the boost direction. k 2 held tensor r.μν-αμα avprop-αμα...
-2, 1), and 4. A is a 2 x 2 matrix with real entries, N(A - 31) = N(A - 1) = c(1,2) for all parts of this problem. (a) (4 points) Is A symmetric? (b) (4 points) Write the solution to the system of differential equations u' (t) = Au(t) if 7(0) = (6,7). (c) (4 points) What is 5e^? Write your answer as a single matrix.
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
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...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
- Derive the equations of motion of the system in terms of variables m and K and express them in matrix notation. Finally, express the equations of motion numerically in matrix notations if the stiffness and mass coefficients are k = 1 kip/in and m = 0.15 kip-sec? / in. Use X1, X2, and X: as degrees of freedom. (20 pts) X2 X 3m