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BiGyElLoWhAt
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Homework Statement
2: Some proofs:
a) If ##\{ v_1 , v_2...v_n \} ## are linearly independent in a real vector space, so are any subset of them.
b) If any subset of vectors ##\{ v_1 , v_2...v_n \} ## in a real vector space are linearly dependent, then the whole set of vectors are linearly dependent.
c) If A is invertible so is A^2
d) if A is not invertible, neither is A^3
I think I either know everything else or I can't really ask without asking to be walked through it so I'll just have to spend some time on google for acouple of them.
Any help is appreciated, my final's in 18 hours.
Homework Equations
The Attempt at a Solution
2:
a)I was thinking I could use closure under scalar multiplication and addition to prove that, but is that solid enough?
If ##\{ v_1 , v_2...v_n \} ## are linearly independent then by definition
##c_1v_1 + ... + c_{n-1}v_{n-1} ≠ c_nv_n##
(is subset ≈ subspace?)
then for ##\{ v_1 , v_2...v_m \} ## where m≤n (not really sure how to notate that all the vectors of this set are contained within the first set) which are all members of the first set, are linearly independent.
This feels weak though... It seems like common sense that if i have some linearly independent vectors, and I throw some of them out, then what I have is still linearly independent; I'm also sure that I need to start at the definition of linearly independent, which basically involves the demonstration of closure under addition and scalar multiplication. I'm just not really sure how to put it technically. I think this is one of those "prove 2+2=4" things and I'm just not seeing how to put it mathematically.
2:
b) Very similar work, I could copy and paste everything I have from up ther and put it here, but once I figure out a, I'll get b.
2:
c) ##A^2 = AA## multiply by ##A^{-1}##
... ##AAA^{-1} = AI = A##
Proved.
I only put this in here because I'm not sure what to do about d. I know it's very similar, but is it simply
##A^3 = AAA## and since A is not invertible, A^3 cannot be reduced? How do I word that?