Answer:
Given that be equipped with the inner product where .
a). Let
now
b). Let and .
will be orthogonal if .
Now
which is not equal to 0.
Therefore are not orthogonal.
3. Let R be equipped with the inner product (x,y) = AX Ay, where A is...
3. Let R3 be equipped with the inner product (x, y) = AX Ay, where A is the matrix shown below: TO -4 21 A = 3 2. LO 0 5) a.) (5 points) Let v = (1,-1,3). Find || V ||. UN b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer
3. Let R3 be equipped with the inner product (x,y) = Ax. Ay, where A is the matrix shown below: TO A=13 LO -4 2 0 2 1 5) a.) (5 points) Let v = (1,-1,3). Find ||v||. b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer.
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...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
3. Let V be the space of n X 1 matrices over C, with the inner product (X\Y) = YGX (where G is an n x n matrix such that this is an inner product). Let A be an n x n matrix and T the linear operator T(X) = AX. Find T*. If Y is a fixed element of V, find the element Z of V which determines the linear functional X + Y*X. In other words, find Z...
Definition 0.1. The inner product is a map <-, - >: V x V +R where V is a vector space satisfying (1) Conjugate symmetry < x, y >= <Y, X > For us in the reals, ignore the complex conjugate. (2) Linearity in the first argument <ax, y >= a< x, y> and < x + y, z >=< 3,2 >+<y, z> (3) Positive definiteness < x, x > 0 and < x, x >= 0 & x=0 The...
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. Let M,, be equipped with the standard inner product. Prove. u is orthogonal W-span w,w,w) 3 -1 note: You must use some of the axioms in the definition of an inner product 2. Let M,, be equipped with the standard inner product. Prove. u is orthogonal W-span w,w,w) 3 -1 note: You must use some of the axioms in the definition of an inner product
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
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...