[Linear Algebra] Nonhomogeneous System Solutions

We learn how to find the solutions of nonhomogeneous systems.

#LinearAlgebra #Algebra #UniversityMath #Lecture

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We discuss solutions of non-homogeneous linear systems and discuss how we write that in parametric form. We also take a look at what it really means to be non-homogeneous.

Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[whiteboycarl]

you saved my life by making these tutorials for linear algebra and discrete maths. thanks!

[alvinag]

Thanks so much for this!! It’s makes so much more sense when you explain it instead of a textbook 😂😂

[thebrainzfx]

your explanations are so intuitive, thank you very much!

[manishbhandariproductions]

you explained it with the visuals so well! Thank you Trev!

[jaysmith]

This seems to be the same thing as the case of an inhomogeneous linear Dif. Eq. where the solution is the general homogeneous solution plus a particular solution to the inhomogeneous equation. I always find it remarkable when the same thing occurs in different branches of mathematics.

[juniorcyans]

If I pick p-q as my homogeneous solution then could I say t(p-q)+p is the non-homogeneous solution of the line M? Or must I set q-p because the line is oriented from p to q?

[maxpercer]

how do you quickly find the 'translation'

[benargee]

What software do you use for your videos? The hand drawing looks so natural.

[TrojenMonkey]

Good explanation, however it feels lacking. You derived the conclusion that Ax = b is built from [homogeneous vector] + [translation], but this is super confusing since you derived that conclusion by row-reducing a nonhomogenous system.
I.E: You explain how to get to M, by starting from 0 -- but in your parametric example, you start from Ax = b, you start from M to find 0 and the translation! This is super confusing, since it doesn't explain WHY after row-reducing a nonhomogeneous system, we end up with a homogeneous vector! I don't understand why inherently row-reducing provides us that?

The only thing I can think of, is the simplification of:
let c and d be solutions to Ax = b, then:
A(c-b) = A(c) - A(b) = b - b = 0.
Let v = (c-b)
As such, A(v) = 0. And every linear transformation of v will also solve this equation, i.e: A(sv) = sA(v) = s0 = 0
So the general solution of the homogeneous system, A(s*v), correlates with the linear transformation of s*c and s*b, two vectors on the same line (m).

But that also doesn't explain why row-reducing a nonhomogeneous system, provides us directly with a [homogeneous vector] + translation !

[pppppp]

can there be a non homogeneous equation with a trivial solution?

[anbazhagane]

can u do a class on DCT, DST, projections

[sushmitanigam]

thanx a lot. sir can u tell me from where to practice such questions to have confidence.??

[eliasoquendo]

For an mxn matrix A, If Ax = 0 has only the trivial solution then Ax = b has a unique solution for each b is an element of the set R^m. Could you explain why this is False apparently that's the answer and I cannot figure out why? Am I trying to devise an example to explain it but I keep getting stuck because if there are pivot positions in every row and Ax = 0 is trivial(no free variables) then Ax = b would be a unique solution. I tried thinking about maybe if Ax = b had more columns then rows but then it wouldn't be trivial.

[steven]

in my solution book, in some cases they use the same variable for two different free variables. for example x3=a and x4=a? why is this?