Assume that xo is a solution to the following linear system Axo where A is an n × m matrix, with n < m, xo is an m-dimensional column vector and b is an n-dimensional column vector. 1. Show that the L² norm of xo can be arbitrarily larger than the norms of A and b. 2. If x- is a vector orthogonal to xo, is xo + x+ also a solution to the above linear system?

Assume that xo is a solution to the following linear system Axo where A is an n × m matrix, with n < m, xo is an m-dimensional column vector and b is an n-dimensional column vector. 1. Show that the L² norm of xo can be arbitrarily larger than the norms of A and b. 2. If x- is a vector orthogonal to xo, is xo + x+ also a solution to the above linear system?

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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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Assume that xo is a solution to the following linear system
Axo
= 6
where A is an n × m matrix, with n < m, xo is an m-dimensional column vector and
b is an n-dimensional column vector.
1. Show that the L² norm of xo can be arbitrarily larger than the norms of A and b.
2. If x- is a vector orthogonal to xo, is xo + x- also a solution to the above linear
system?

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