2) The set S of all real-valued functions f(x) of a single real variable z is...
2) Let CI0,1] be the vector space of all continuous real valued functions with domain [0,1J.Let (f.8)-Co)ds be the inner product in C10.11 where fand g are two functions in CI0,1. Answer the following questions for f(x)-x and g(x)-cos. a) Find 《f4) and i g I where l.l denotes the length induced by this inner product,Show your work b) Determine the scalar c so that f-cg is orthogonal to f.Show all your work.
NEED (B) AND (C) 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval -00 < x < oo that approach zero at least as quickly as xas +00 (a) (4 points) Show that the operator B=x+in is Hermitian when acting upon {f(x)}. (b) (4 points) Show that A = x + (4) is not Hermitian and determine the operator At. Determine C = AtA and show that it is Hermitian. (c) (2 points) What well-known problem in...
advanced linear algebra, need full proof thanks Let V be an inner product space (real or complex, possibly infinite-dimensional). Let {v1, . . . , vn} be an orthonormal set of vectors. 4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
Let f(x) and g(x) be any two functions from the vector space, C[-1,1] (the set of all continuous functions defined on the closed interval [-1,1]). Define the inner product <f(x), g(x) >= x)g(x) dx Find <f(x), g(x) > when f(x) = 1 – x2 and g(x) = x - 1
# 4: For smooth complex valued functions f(x), g(z) defined for 0 < x inner product<f(x),g(x) > by 2π define the Hermitian Introduce the operator D(f() a)Show that <D(f(x),9()), D(g(x)) > if f b) For n and integer show that einz for 0-x-2n satisfi c) Show that for mメn both integers then < einz, enny-0, 0,警) (0)- ic boundary conditions. Also onormal and < einz, einz >-2T. θ, Call these last periodic boundary conditions for f(x), g(s), show that D(einz)...
Let F be the set of all real-valued functions having as domain the set R of all real numbers. Example 2.7 defined the binary operations +- and oon F. In Exercises 29 through 35, either prove the given statement or give a counterexample. 29. Function addition + on F is associative. 30. Function subtraction - on is commutative
33. Let C[0, 1] be the space of real-valued continuous functions on [0, 1] with inner product Kf.9) (x)g()d(r). 2 cos 2Tir and g.(r)-v2 sin 2mix for i 1.2,... Show that (1. fi.g. f2 92 Suppose that fi(r) is an orthonormal set.
(4) Let C[0,1] be the inner produce space of all real-valued, continuous functions on the interval (0,1) with inner product.g) = Sopr)(x) dr. Determine the projection of the vector {m} onto the space spanned by the orthonormal system of vectors given below. {1, 73(2x - 1)}
Please answer C 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...