Homework Answers
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R sat...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R satisfying the following propertics The map g is bilinear. That is, for all v, v1, V2, w, w1, W2 CV and all t1,2 ER The map g is symmetric. That is, g(v, w) g(w, v) for all v, weV. The map g is positive definite. That is, g(v,v) 0 for a v e V with equality...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a sequence of distinct vertices V-(vi,v2, ,%)-V connected by a sequence of edges {(vi, t), (Ug, ta), , (4-1,Un)) : We say that (V, E) is a path from tovn. Sketch a graph with 10 vertices and a path consisting of 5 vertices and four edges. Formulate a binary integer program that could be used to find the path of least total weight from one...
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W...
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W of R" and {...., } be an orthogonal basis for Wt. Determine which of the following is false. a. p+q=n b. {U1,..., Up, V1,...,0} is an orthogonal basis for R". c. the orthogonal projection of the u; onto W is 0. d. the orthogonal projection of the vi onto W is 0. e. none of these 8. Let {u},..., up} be an orthogonal basis...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
Please give me right answer with the details. Thanks! Let Vi, V2,. .., V, be mutually...
Please give me right answer with the details. Thanks!
Let Vi, V2,. .., V, be mutually orthogonal subspaces of R". In other words, if ij, then vlw for all ve V,, w E V;. Prove that dim Vi dim V2 + .. + dim V, < n.
Let Vi, V2,. .., V, be mutually orthogonal subspaces of R". In other words, if ij, then vlw for all ve V,, w E V;. Prove that dim Vi dim V2 + .....
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i},...
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...
Let G = (V. E) be an undirected, connected graph with weight function w : E → R
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the max...
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the maximuin value of G(w) b - [2 pts| Derive explicitly the impulse response of the following system n] Hint: Besides some graphical consideration, there is no calculation. The answer is mostly based orn the use of properties. c - 3 pts] Up to...
Let {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Ho...
Let {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Hom(V, W ) s.t. ∀ i ∈ {1, 2, ..., n}, α(vi) = wi
Let G = (V, E, W) be a connected weighted graph where each edge e has...
Let G = (V, E, W) be a connected weighted graph where each edge e has an associated non-negative weight w(e). We call a subset of edges F subset of E unseparating if the graph G' = (V, E\F) is connected. This means that if you remove all of the edges F from the original edge set, this new graph is still connected. For a set of edges E' subset of E the weight of the set is just the...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R satisfying the following propertics The map g is bilinear. That is, for all v, v1, V2, w, w1, W2 CV and all t1,2 ER The map g is symmetric. That is, g(v, w) g(w, v) for all v, weV. The map g is positive definite. That is, g(v,v) 0 for a v e V with equality...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a sequence of distinct vertices V-(vi,v2, ,%)-V connected by a sequence of edges {(vi, t), (Ug, ta), , (4-1,Un)) : We say that (V, E) is a path from tovn. Sketch a graph with 10 vertices and a path consisting of 5 vertices and four edges. Formulate a binary integer program that could be used to find the path of least total weight from one...
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W of R" and {...., } be an orthogonal basis for Wt. Determine which of the following is false. a. p+q=n b. {U1,..., Up, V1,...,0} is an orthogonal basis for R". c. the orthogonal projection of the u; onto W is 0. d. the orthogonal projection of the vi onto W is 0. e. none of these 8. Let {u},..., up} be an orthogonal basis...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
Please give me right answer with the details. Thanks!
Let Vi, V2,. .., V, be mutually orthogonal subspaces of R". In other words, if ij, then vlw for all ve V,, w E V;. Prove that dim Vi dim V2 + .. + dim V, < n.
Let Vi, V2,. .., V, be mutually orthogonal subspaces of R". In other words, if ij, then vlw for all ve V,, w E V;. Prove that dim Vi dim V2 + .....
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...
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the maximuin value of G(w) b - [2 pts| Derive explicitly the impulse response of the following system n] Hint: Besides some graphical consideration, there is no calculation. The answer is mostly based orn the use of properties. c - 3 pts] Up to...
Let G = (V, E, W) be a connected weighted graph where each edge e has an associated non-negative weight w(e). We call a subset of edges F subset of E unseparating if the graph G' = (V, E\F) is connected. This means that if you remove all of the edges F from the original edge set, this new graph is still connected. For a set of edges E' subset of E the weight of the set is just the...
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