5.4.17. Construct an example using the standard inner product in R to show that two vectors...
4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis (b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x)on the interval [0, 1. Hint: You may use the following result without proof f Ine* dr = (-1)"(ane-n!), where ao = 1, an- | n. + | , for n-1, 2, ). 4) Consider the inner product space P2(R),...
(4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis 11, r, r2) b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x) e on the interval [0, Hint: You may use the following result without proof: J İlne dra(-1)"(ane-n!), where ao-1, an-le! + îl , for n-1, 2, or n=1,2 .. ). (4) Consider the inner product space...
5. (a) Explain why the standard inner product is invariant under an orthogonal trans formation. That is, if U is any orthogonal miatrix, and if u = Ux and v = Uy, then i.e. multiplication by an orthogonal matrix does not change the standard inner product. (b) Given any two vectors x. y in R", explain why the angle between them is Py invarient under an orthogonal transformation. That is, if u where P is an orthogonal matrix, thern Px...
5.4.3. Consider the following set of three vectors. X2? 0 (a) Using the standard inner product in 4, verify that these vec tors are mutually orthogonal (b) Find a nonzero vector x4 such that (x1, x2, x3, x4) is a set of mutually orthogonal vectors. c) Convert the resulting set into an orthonormal basis for .
4) Consider the inner product space P2(R), with inner product 0 (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis (b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x)on the interval [0, 1 (Hint: You may use the following result without proof: oe d(an!)where a 1, anor n1,2....) ane- n!), where do -I, ln
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
When dealing with standard vectors (with purely real elements) we are used to finding the angle between the vector from But what happens when we are dealing with vectors that have complex elements. In quantum mechanics, in general, the inner product is a complex number, which does not define a real angle The Schwarz Inequality helps us in this regard However, according to it, the only thing we can know is that the absolute value of the inner product is...
6. 2D vectors Lec ture Supplement 4: Intro Vectors Worksheet B Provide an example of a) ID b) 2D c) 3D a vector (graphical, verbal, or mathematical) that is in: (graphi Outline the main vector operations we will use in class: a) Vector Addition b) Vector Subtraction c) Scalar Multiplication d) Vector Dot Product e) Vector Cross Product What is a resultant vector? 4 What is the component of a vector? &Define a unit vector. Give an example of a...
Question 4) (6 points) Below are two unrelated questions that both deal with inner products. Note part (a) does not connect to part (b) (a) Consider the inner product on PX(R) defined by < p(x), g(x) >= } p(x)q(z)dir Show that the vectors x2 and 42 - 3 are orthogonal in this inner product space. (b) Conisder the vector space M.(R). Give an example to show why the following is not a valid inner product on this space: <A, B...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...