4.11 Let )s F 2n F2n F be defined as (u, v), (u', v)s u .v -vu where u, v, u', v' e F and is the Euclide...
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i}, 10. (7 points) Given a where d(u, v) is the length of a shortest path between u and v. (a) (0.5 point) What is B1(G)? (b) (1.5 points) Let Pn be the path with n vertices. What is B;(Pn)? (c) (2 points) Show that if G is an n-vertex 3-regular graph, then B2(G) < . Further- more, find a 3-regular graph H such that...
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such
(9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product on V if and only if T is positive (per our definition of positivity.
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Let V be a real inner product space. Under what condition on u, v E V is the following equality valid: where l 1x12 = (x, x) YE V.
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
4.12 For (u, v) F, symplectic weight wts((u, v)) of (u, v) is defined to be the number of 1sin such that at least one of ui, v is nonzero. Show that where u = = (u1,. , u) and v (1, ), the wt((u, v))wts(u, v))w((u, v) where wt((u, v)) denotes the usual Hamming weight of (u, v)
4.12 For (u, v) F, symplectic weight wts((u, v)) of (u, v) is defined to be the number of 1sin such...
you can skip #2
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u where f(r,y,) = =- +22 2. Consider the vector field F(E,) = (a,y) Compute the flow lines for this vector field. 3. Compute the divergence and curl of the following vector field: F(x,y,)(+ yz, ryz, ry + 2)
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u...
Let W(s, t) - F(u(s, t), vis, t)), where F, u, and v are differentiable, and the following applies. u(6, -6) - 7 v(6, -6) -9 us(6, -6) - 2 vs(6, -6) -7 (6,-6) --4 V:(6, -6) = 3 Fu(7.-9) - - 1 F (7.-9) - -2 Find W (6, -6) and W.(6, -6). Ws(6, -6) W:(6, -6) =