1. If the vectors and are orthogonal with respect to the weighted inner product
< > = , what must be true about the weights ?
2. Do there exist scalars k and m such that the vectors p1 = 2+kx+6, p2 = m+5x+3 and
p3 = 1 + 2x + 3 are mutually orthogonal with respect to the standard inner product on P2?
1.
<u,v> = w1•u1•v1 + w2•u2•v2 = w1•1•2 + w2•2•(-4) = 2•w1 - 8•w2
If, u & v are orthogonal , then, <u,v> = 0
So, 2•w1 - 8•w2 = 0
So, w1 = 4•w2 is the relation between w1 & w2 in order that u & v are orthogonal.
1. If the vectors and are orthogonal with respect to the weighted inner product < >...
Let V be a finite dimensional inner product space, w1,w2V. Let TL(V) and Tv=<v,w1>w2 for all vV. Find all eigenvalues and the corresponding eigenspaces of T. Please provide full solution. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
In Exercises 78-79, the given vectors are respect to the Euclidean inner product. Find proj (1, 2, 0, -2) and W is the subspace of R4 spanned by the orthogonal with X, where vectors. v Stios 78. (a) Vi (1, 1, 1, 1), v2 = (-1, -1, -1, 1) (b) vi = (1,0, -3, -1), v2 = (4, 2, 1, 1) In Exercises 78-79, the given vectors are respect to the Euclidean inner product. Find proj (1, 2, 0, -2)...
Let be an inner product space (over or ), and . Prove that is an eigenvalue of if and only if (the conjugate of ) is an eigenvalue of (the adjoint of ). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageTEL(V) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
(3 points) The function (f, g) = f(-2x(-2) + f(0)g(0) + f(2)g(2) defines an inner product on P2. With respect to this inner product, find the orthogonal projection of /)-4x2 +5x-3 onto the subspace L spanned by g(x) = 2x2-2x-4.
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1. Let be the operator on whose matrix with respect to the standard basis is . a) Verify the result of proof " is normal if and only if for all " for question 1. Note: stands for adjoint b) Verify the result of proof "Orthogonal eigenvectors for normal operators" for question 1. The proof states suppose is normal then eigenvectors of corresponding to distinct eigenvalues are orthogonal. We were unable to transcribe this imageWe were unable to transcribe this...
Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13 Let P2 have the inner product given by evaluation at -5, -1, 1, and 5. Let po(t) = 2, P1(t)=t, and q(t) = 12 approximation to p(t) = t by polynomials in Span{Po.P1,9}. The best approximation to p(t) = t by polynomials in Span{Po.P2,q} is
Ch6 Inner-product and Orthogonality: Problem 14 Previous Problem Problem List Next Problem (1 point) All vectors are in R". Check the true statements below: A. Not every linearly independent set in R" is an orthogonal set B. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. C. A matrix with orthonormal columns is an orthogonal matrix. D. If L is a line through 0 and itỷ is...
1. (Difficulty: *) Write the value for the inner product (v(0), v(1)) where VO) and y(1) Enter answer here 2. (Difficulty: **) Consider the following vectors in R4 v (0) (1) and (2) = You can verify that the vectors are mutually orthogonal and have unit norm. How many different vectors v3) could we find such that {v(©), v(1), v(2), v(3)} is a full orthogonal basis in R4? 0 1 2 3 >3
(2) Use projections onto the 8 basis vectors you got in (1) to decomposo the vector xint parallel and orthogonal components tospan(b) We were unable to transcribe this image (2) Use projections onto the 8 basis vectors you got in (1) to decomposo the vector xint parallel and orthogonal components tospan(b)