Let v and w be vectors in an inner product space V. Show that v is...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
8. Show that if [u, v, w\ is an orthogonal set in an inner product space V, then
4. Let TV - V be a linear operator on a finite dimensional inner product space V and P be the orthogonal projection of V onto the subspace W of V. a) Show that is invariant under T if and only if PTP = TP. b) Show that w and we are both invariant under 7 If and only if PT = TP
QUESTION 8 Let (V,<,>) be an inner product space, and P: V – V a linear map. Choose the correct statement(s). Multiple choices might be correct and wrong choices have negative points. if P(V) = < W, V > Wand ||w|= 1, then P is an orthogonal projection. if P is an orthogonal projection, then < V- P(V), W> = 0 for any VEV, welmP. fW= Im P and {W 1,...,Wx} is an orthonormal basis for W then P(V) =...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
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). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
2. Suppose that V is an inner product space. (i) Prove that, for any vectors 01, 02 € V, || 0111? + || 0,2||2 = || v1 + v2||2 + || 01 – v2||2 2 (ii) Prove that, for any vectors V1, V2 € V, if v, and v, are orthogonal then || 01 || + || 112 || 2 = || 01 + 02||2.
b) Let V be a complex vector space, let (,) be an inner product on V, and let 2, y E V be certain vectors. Assume that (x, y) = 2i and (y, y) = 5. Find (< + iy, iy).
Let (V, (,) be an inner product space and T:V → V and S:V + V be self adjoint linear transformations. Show that TOS:V + V is self adjoint if and only if SoT=TOS.