Finding a Solution to a Complex Homework Problem

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Homework Help Overview

The discussion revolves around a linear algebra problem involving a zero matrix A and a set of vectors {X1, X2, ... ,Xq}. The original poster is tasked with proving that AXp is not zero for some p, despite A being a zero matrix. Participants explore the implications of the span of the vectors and their relationship to the matrix A.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the vectors and the matrix A, questioning the implications of the span and the null space. There are attempts to clarify the definitions of R^n and R^m, as well as the nature of matrix multiplication. Some participants suggest that certain vectors in R^n could yield a non-zero result when multiplied by A, while others challenge these assumptions.

Discussion Status

The discussion is active, with participants providing hints and guidance on visualizing matrix multiplication and understanding the implications of the span. There is recognition that not all vectors will yield a non-zero product with A, but some participants are beginning to see the connections between the vectors and the matrix.

Contextual Notes

There is some confusion regarding the definitions and properties of spans and linear combinations, as well as the specific requirements of the homework question. Participants are working through these concepts while adhering to the constraints of the problem.

  • #31
Aah Ax is non zero! for some reason I was thinking Ax was equal to zero.

So instead a1*(A*X1)+...+aq*(A*Xq) is NOT equal to zero, meaning the A*Xis can't be equal to zero?
 
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  • #32
Right. SOME of the A*X's can be zero. But certainly not ALL of them. We PICKED x so that Ax was nonzero, right? Whew! Thanks for playing along.
 
  • #33
so when the question says to prove that A*Xi is not equal to zero, that doesn't hold for EVERY Xi in the span? the way it is phrased makes it sound like I'm supposed to show that it's always true that A*Xi is not 0, but if Xi = 0 that would be untrue.

is that the end of the proof then?
 
  • #34
The question says "Prove that AXi is NOT zero for some i.". I just looked it up and quoted it exactly. Wouldn't you agree that's the same thing as proving that "Not all of AXi are equal to zero"? The proof's not over until you say "I understand" and can repeat it. You don't have to repeat it here. Just say you can.
 
  • #35
Yes I suppose that does make sense. I was thinking 'some i' was synonymous with 'all i'. I am pretty new to proofs and math that doesn't use numbers, as you can probably (certainly) tell, hah.

And yes, I could repeat it now. I'm not sure I could do other similar proofs on my own yet though... I suppose I'll have to practice a whole bunch. and get a tutor.

Anyway, 2 and a half hours later - thank you so so much! you have been a great help.
 
  • #36
I think you've picked up some good points here and I think you'll get better. Thinking 'some i' is the same as 'all i' is pretty typical of the problems you are having. These words aren't interchangable. If you do choose to get a tutor, ask for help on basic logic in proofs, ok? The language isn't as vague as you are treating it.
 
  • #37
I really hope this is one of those things that gets easier as you do more problems. I will definitely mention basic logic to my tutor if I get one. I tried searching it on the internet but didn't find much. Thanks again for the help and advice!
 

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