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Let A € Rnxn. Prove or disprove that Ax = 0 and AA x = 0 have the same solution sets.

Let A € Rnxn. Prove or disprove that Ax = 0 and AA x = 0 have the same solution sets.

Advanced Engineering Mathematics
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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(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.
Transcribed Image Text:(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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