- #1
Why are the columns of Q linearly independent?
- Thread starter victoranderson
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- Independent Linearly Matrix
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Homework Help Overview
The discussion revolves around the linear independence of the columns of a matrix Q in the context of matrix transformations and properties. Participants are exploring the relationship between matrices and their columns, particularly in relation to a matrix M and its transformations.
Discussion Character
- Conceptual clarification, Mathematical reasoning, Problem interpretation
Approaches and Questions Raised
- Participants are attempting to understand the derivation of the columns of Q and their relationship to the matrix M. Questions are raised about the methods to determine these columns without relying on trial and error. There is also exploration of the implications of matrix multiplication and the conditions for linear independence.
Discussion Status
The discussion is active, with participants sharing insights and questioning each other's reasoning. Some guidance is offered regarding the relationships between the matrices, but there is no explicit consensus on the methods or conclusions regarding the linear independence of the columns of Q.
Contextual Notes
Participants express uncertainty about their understanding of the matrix relationships and the implications of their findings. There is mention of specific column vectors and their transformations, but the overall context remains complex and not fully resolved.
- #2
tiny-tim
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hi victoranderson! 
if Q-1MQ = that matrix,
then MQ = Q(that matrix), which is … ?
then MQ = Q(that matrix), which is … ?
- #3
victoranderson
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tiny-tim said:hi victoranderson!
if Q-1MQ = that matrix,
then MQ = Q(that matrix), which is … ?
I know what you mean
Let that matrix be D
I can find MQ=QD by trail and error
Is there any other method I can use to find the columns of Q without trial and error?
- #4
tiny-tim
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hi victoranderson!
(just got up :zzz:)
it isn't trial and error
what are the columns of QD ?
(just got up :zzz:)
victoranderson said:
it isn't trial and error
what are the columns of QD ?
- #5
victoranderson
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tiny-tim said:hi victoranderson!
(just got up :zzz:)it isn't trial and error
what are the columns of QD ?
it's been a long time..
The columns of QD is [0,0,0]^T, [1,1,-1]^T and [2,3,1]^T
which are ##M^3v , M^2v , Mv ## respectively
I am stupid and I still do not understand..
Last edited:
- #6
tiny-tim
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hi victoranderson!
for any matrix P with columns A B C:
so MP = PD (ie P-1MP = D) if 0 = MA, A = MB, B = MC
(so A = MB = M2C and M3C = 0)
so we choose the columns of P to be M2 MC and C
(and then we call it Q instead of P)
for any matrix P with columns A B C:
the columns of PD are 0 A B
the columns of MP are MA MB MC
the columns of MP are MA MB MC
so MP = PD (ie P-1MP = D) if 0 = MA, A = MB, B = MC
(so A = MB = M2C and M3C = 0)
so we choose the columns of P to be M2 MC and C
(and then we call it Q instead of P)