Denote by Mmxn (F) the vector space over F of all (m × n) matrices with entries in F. For A €Mmxn (F), we usually write A = [aij] where aij E F are the entries of A with i = 1, 2, ..., m andj = 1, 2, n. Let2a: M3x3 (R) → Rbe defined by3 3a([aij]) = ΣΣ dij.i=1 j=1Find a basis for the kernel of a.

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Answered: Denote by Mmxn (F) 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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Consider the matrix V E R"xn where V has the structure V = I – 2vvT, the identity matrix I E R"Xn, and the vector v E R". The matrix V depends only on the identity matrix and vector v # 0, and occurs frequently in scientific computing and optimization applications. Task: Can you design an orthogonal matrix V? That is, if possible, pose a condition on the vector v / what property does v need to have, such that V = I – 2vvT is an orthogonal matrix? (v cannot be the zero vector, v # 0)
Let J, be then X n matrix whose entries are all equal to 1, and let D(^1, . , An) be the n X n diagonal matrix whose non-zero entries are 11, ... , An E R. Let x = (x1, ... , Xn) be a row vector in R' and let x' be its transpose, a column vector. 1. Show that xJ,x' > 0, for all x1, ... , Xn E R. 2. Assume that the scalars 11, ..., An are all strictly positive. Show that x D(11, ... , An)x' is strictly positive for all (x1, ..., Xn), unless x1 = •… Xn = 0. 3. Use questions 2 and 1 above to show that D(^1, .. , An) + 1Jn is an invertible matrix when 11, , ^n and 1 are all strictly positive.
(a) Let A € Rnxn. Prove or disprove that Ax = 0 and AAT x = 0 have the same solution sets. (b) Let & € Rx1 be a unit vector and I be the n x n identity matrix. Prove that det (IT) = = 0. (c) Consider the quadratic form f(x1, x2, x3) = 2(a₁x₁ + a2x2 + A3x3)² + 3(b₁x₁ + b₂x2 + b3x3)². (i) Let a = a1 az , B = b₁ b₂ b3 Show that the matrix of the quadratic form is 2aa¹ +388¹. (ii) Show that if a and 3 are unit vectors and are orthogonal to each other, then there is an orthogonal change of variable 8-8 P X2 Y2 Y3 so that f(x1, x2, x3) can be written as 2y² + 3y2.
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