Let \lambda_1,\lambda_2,\lambda_3,b\in \mathbb{R} with \lambda_1\neq0. Show that the linear equation

Cosmo Truong

Cosmo Truong

Answered question

2022-02-17

Let λ1,λ2,λ3,bR with λ10. Show that the linear equation λ1x1+λ2x2+λ3x3=b has a set of solutions of the form {v}+span(v1,v2), with v,v1,v2R3 and v1,v2 are linearly independent.
Could you give me a hint how we could show that, since I have no idea? I don't really know how to start.

Answer & Explanation

Elodie Williamson

Elodie Williamson

Beginner2022-02-18Added 7 answers

Note that the rank of the system is one (since λ10) and then the null space has dimension 2(=3-1).
To find the solution note that
v=(bλ1,0,0)
is a particular solution and
v1=(λ2,λ1,0)
v2=(λ3,0,λ1)
are two linearly independent solutions to
λ1x1+λ2x2+λ3x3=0 therefore all the solutions for
(x1,x2,x3) are given by
{v}+span(v1,v2)

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