Given: vector space R3 and the function: 〈x, y〉= 2x1y1 + x2y2 + 5x3y3
a) Show that the given function defines an inner product on R3.
b)
c)
We need at least 9 more requests to produce the answer.
1 / 10 have requested this problem solution
The more requests, the faster the answer.
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3 -> R3 that to each vector ~v assigns the vector of S given by f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a linear function. (iii) What is the matrix of f in the standard basis of R3? (iv) What are the null space and the column space of the matrix that you computed in the previous point? Exercise 1. In...
c and d only 2. Consider the vector space R3 with the standard inner product and the standard norm |x| x, x) Use the formula for projection given in Chapter 5, Section 4.2 of LADW to find the matrix of orthogonal projection P onto the column space of the matrix -) 1 1 A = 2 4 (a) What is the projection matrix P? (b) What is the size of P? (c) Since the dimension of the column space of...
2.4. Let V be a vector space and let vi,V,..., Vn be a basis in V. For x Prove that (x, y) defines an inner product in V
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): -1 1 ( 2 5 3 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, g(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2x2 matrices: (You'd decided what the inner product was on a previous math...
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B. 4. Consider the vector space...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements 14. Let Ps denote the vector space of polynomials of degree 3 or less,...