Problem 5.Consider the vector space R4. Find all pairwise orthogonalvectors (column vectors) x1,..,Xp, where the entries of the column vectors canonly be +1 or –1. Calculate the matrixj=1and find the eigenvalues and eigenvectors of this matrix.

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Answered: Problem 5. Consider the vector space…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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13. [R For each of the following 2 × 2 matrices, draw a picture to show Ae1, Ae2, Ab, where e and e, are the standard basis vectors in R? and where b = 2ej + 3e2. w 6 ) » (). (9), (3) () -2 0 0 2 -2 -2 4 -4 ), d) 0.7 0 -3 -1 -4
Find a and that solve the vector equation: a) 1 13 a (²) + ² ( ₂ ) = (₂3) 3 29 Suppose the above equation is expressed as Ax = b with x = the coeffieints of the equations in one matrix form as (Ab)= Rows: 2 Columns: 3 (²). Express
Solve the question below:
I got some answers for there questions but not sure if they are right
In this exercise you will explore the ideas of orthogonal vectors, orthogonal matrices, and why these ideas are useful: (a) Plot the vectors v₁ = (1) in the xy-plane. 1 (b) Compute v₁ V₁ and v₂ v₂. The fact that you get an answer of 1 for both is due to the length of the vectors being equal to 1. Now compute v₁ · v₂ and v2 - V₁. The fact that you get an answer of 0 for both of these answers is due to the fact that these two vectors are perpendicular or "orthogonal" to one another. (V1 V₂) (1 whose columns are v₁ and v₁ respectively and the matrix , V2 = (c) Now consider the matrix A = = √/2 = V1 ᎪᎢ √2 V₂ A AT and AT A. What does this tell you in terms of inverse matrices? (d) What do you think is the general fact one can say about how to construct these nice “orthogonal matrices" like A in general? How would you go about it (in words, don't actually come up with an example) for a 3 × 3 matrix? whose rows are v₁ and v₂ transposed (i.e. turned sideways) respectively. Compute
Q2. Let A = (aj) be a 10 × 10 positive definite matrix with Cholesky factor R = (rij). Suppose that a11 = 9, 912 12, 017 = 3, a22= 25, and a27= 7. Solve for 727.
Problem 6. Prove that the vectors v₁ = (1, 0, -1), U₂ = (1, 2, 1), V3 = (0, -3,2) form a basis for R³. Express each of the standard basis vectors, (1, 0, 0), (0, 1, 0), and (0,0,1) as a linear combination of v₁, U2, U3.
Qa) attached below Qb) attached below Qc) ? (alpha) = ? Qd) ? (beta) = ?
Please solve as soon as possible
3/20 sin x - X dx = ? 22m (a) -0". (2n) (2n)! 22n+2 (b) E(-1" n=1 (2n + 2) (2n + 2)! n-1 22n11 (2n +1).(2n+1)! 22n 1 (2n-1) (2n-1) n=1 n=1 2" (e) > n.n! n=1
Please solve part d only

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Problem 5. Consider the vector space R4. Find all pairwise orthogonal vectors (column vectors) x1, ...,Xp, where the entries of the column vectors can only be +1 or –1. Calculate the matrix j=1 and find the eigenvalues and eigenvectors of this matrix.