Answered: Show that the solution set to a system…

Show that the solution set to a system of equations of the form au*, + Azyš, + + a, x = 0 Inn + ar = 0 amx, +...+ a_x_ = 0, where the a's are real, is a subspace of R".

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.5: Subspaces, Basis, Dimension, And Rank

Problem 10EQ

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Question

Show that the solution set to a system of equations of the form
au*, +
Azyš, +
+ a, x = 0
Inn
+ ar = 0
amx, +...+ a_x_ = 0,
where the a's are real, is a subspace of R".

Expert Solution

Step 1

The given system of homogeneous linear equations is equivalent to a matrix equation of the form:

Ax = 0

Step 2

Clearly x = 0 is a solution.

A0 = 0 ⇒ 0 is a solution ⇒ solution set is not empty

If Ax = 0 and Ay = 0 then A(x + y) = Ax + Ay = 0

If Ax = 0 then A(rx) = r(Ax) = 0.

Step by step

Solved in 3 steps

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