Exercise 2. Given a permutation o E S. define a matrix P, E M. (F) by...
Please solve all in detail! 7. Recall that if o E Sn then P, is the nxn "permutation matrix” satisfying Col;(P.) = Co(j) for j = 1,...,n. (See the Tutorial notes for Feb. 24 for more information.) (a) Prove that if o, T E Sn then P.P, = Por. (Hint: it suffices to prove Col;(P.P.) = Col;(Por) for all j. Use the general fact that Col;(A) = Ae..] (b) Suppose o E Sn given in cycle notation is o =...
1: 1 131 2 Given matrix A 2 2 2. matrix P and I S set 2. a) Show that matrix P diaqonalizes A and find D(diagonal matnx) that matches. 6) Find the eigen values of A Observe that the columns of P form set S c) orthogonal Set using the inner product standard show that set S is not an Use the Gram- Schmidt process to get an orthonormal set from S using inner product standard 1: 1 131...
l. (20 pulita total) Given is A-| 1 3 . Tins matrix has eigruvalunAi-Ag-2 and la-s. (a) (2 points) Explain wby the matrix A can be orthogonally dingoaleed (b) (4 points) Show that i is a l-for the eigenspace eurrespcoding to 0 Xi- "xz- -2 (c) (3 points) Show thatxis a basis for the eigeneqpace responding to A (d) (5 points) plc by an orthoyonal basis Hint: This can be dose by applying the Gra- by Schmidt process to (x....
6. [20 pts.] Below is the final P matrix after applying Floyd's all pairs shortest path algorith on a graph with nodes (A, B, C, D, E, F, G, H). In the matrix below 1 corresponds o 0 5 0 2 0 5 5 Determine the shortest path between nodes D and F. a) 6. [20 pts.] Below is the final P matrix after applying Floyd's all pairs shortest path algorith on a graph with nodes (A, B, C, D,...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...