5. Let S = {vi, u2, , v) be a set of k vectors in Rn with k > n. Show that S cannot be a basis for Rn.
5.) For the integral S(8 - x) dx (a) Show to construct Rn (right hand Riemann sum with n sub intervals) (b) Simplify Rn using your sigma skillz (c) Take limit of R, as n → to evaluate given integral (d) Compute given integral by FTC to check answer
Let V be Rn with a basis B={b1,. bn); let W be Rn with the standard basis, denoted here by E and consider the identity transformation I VW, where l(x) x. Find the matrix for I relative to s and E. What was this matrix called in Section?
Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi, . . . , w, є Rn. If wi є span {v1, . . . , vr} for each i = 1, . . . , s, then spanfVi, . .., v-) -spanfvi, . .., Vr, W,...,w,) Suggestiorn: To see how the proof should go, first try the case s - 1, r 2..]
Problem...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
14. Let S by any set in RN. Let C consist of all convex combinations 04x4 + ... + 0,** with 0; 20, 20; = 1, x'ES. The set C is called the convex hull of S. Prove that C is convex.
4. In lectures, we defined closed subsets of Rn. The definition can be generalized in the following way. Let X be a subset of R". We say that a subset S C X is closed in X if all limit points of S that are in X are also in S. [Any closed subset of Rn is "closed in Rn*) State whether each of the following sets S is closed in X. For cases where X - Rn (including the...
7) Let O S Rn be open and suppose f : O → R is differentiable on O. Suppose has a local maximum or minimum at zo E O. Prove that f'(zo) = 0.
7) Let O S Rn be open and suppose f : O → R is differentiable on O. Suppose has a local maximum or minimum at zo E O. Prove that f'(zo) = 0.
n = 0.0016 , E = 0.005 m [Rn] s. 2 n OD 18 у 2m The asphalt-lived trapezoidal channel in Fig. carries 10 m/s of water under the uniform flow condition when So 0.0015 what is the normal depth ?