Exercise 6 subspace S of V. Show that lIpll2 (p, ) Exercise 6 subspace S of V. Show that lIpll2 (p, )
Exercise 6 (6.4.35, p.452) Let A e Cnxn, and let S be a k-dimensional subspace of C". Then a vector ve S is called a Ritz vector of A from S if and only if there is a pie C such that the Rayleigh-Ritz-Galerkin condition Av – uv Is holds, that is, (Av – uv, s) = 0 for all s E S. The scalar u is called the Ritz value of A associated with v. Let 91, ...,qk be...
I need help on #5 and #7
Exercise 3.5. Let p: G the fixed subspace GL(V) be a representation of a finite group G. Define Vα εV | 99υ = υ,Vgε G. 1. Show that ve is a G-invariant subspace. 2. Show that 1 ΕΣ ε να |G hEG for all v e V. 3. Show that if v E V, then 1 Σ \GI hEG 4. Conclude dim VG is the rank of the operator 1 P = |G|...
Exercise 4.10.47 Consider the set of vectors S given by S -{I 4u+v-5w 12u+6 - 6 4u+4v+4w : U, V, W ER Is S a subspace of R3? If so, explain why, give a basis for the subspace and find its dimension.
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y and x -y are eigenvectors of A. What are their corresponding eigenvalues? (ii) Show that 0 is an eigenvalue of R" with n - 2 linearly independent eigenvectors. (iii) Explain why A is diagonalizable.
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y...
True or False If V is a vector space and S is a subspace of V, then the span of S is the same as S.
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the rank of A is 2 (iii Show that x and y are eigenvectors of A for an eigenvalue X of A. What is A?
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the...
6. (10) Show that if W is a k-dimensional subspace of an inner product space V (not necessarily finite dimensional), then b - projwb is perpendicular to every vector in W. Here projwb is the orthogonal projection of b onto W. (Hint: Use the theorem that W has an orthonormal basis (a, a, .., ak), show that (b - projwbla) = 0, for all :)
1. Show that SCR, S o is a subspace if and only if it is closed under taking linear combinations Civit... + V V ES, C, E R. Hint: For one direction use induction on n.
6. For the following vector spaces V, determine if the subset H is a subspace. If not, give one reason why H fails to be a subspace. (a) (5 points) V is the set of functions f from R + R, and H is the set of polynomials of integer coefficients. (b) (5 points) V = P, is the vector space of polynomials of degree at most 2, and H is the subset of all polynomials in P2 of the...
(1 point) Determine whether the given set S is a subspace of the vector space V. A. V = R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed mxn matrix. B. V is the vector space of all real-valued functions defined on the interval (-oo, oo), and S is the subset of V consisting of those functions satisfying f(0) 0 C. V Mn (R), and S is the...