Finding a Solution to a Complex Homework Problem

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SUMMARY

The forum discussion revolves around a complex linear algebra homework problem involving a zero matrix A and the span of vectors {X1, X2, ..., Xq}. The main conclusion is that if A is a non-zero matrix, there exists at least one vector in R^n such that AXi is non-zero for some i. The participants clarify the relationship between the span of vectors and the null space of A, emphasizing that the span of the X vectors must cover R^n, thus ensuring that not all AXi can equal zero. The discussion highlights the importance of understanding linear combinations and the implications of matrix multiplication.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically matrix multiplication.
  • Familiarity with vector spaces, particularly R^n and R^m.
  • Knowledge of the concept of span in linear algebra.
  • Basic proof techniques in mathematics.
NEXT STEPS
  • Study the properties of linear transformations and their effects on vector spaces.
  • Learn about the null space and range of matrices in linear algebra.
  • Explore the concept of linear independence and dependence among vectors.
  • Practice constructing and understanding proofs in linear algebra.
USEFUL FOR

Students studying linear algebra, particularly those tackling proofs involving matrix operations and vector spaces. This discussion is beneficial for anyone seeking to deepen their understanding of the relationships between matrices and their corresponding vector spaces.

  • #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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