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Why are the columns of Q linearly independent?
- Thread starter victoranderson
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In summary, the conversation discusses finding the columns of matrix QD without trial and error. It is explained that the columns of PD are 0, A, and B, and the columns of MP are MA, MB, and MC. By choosing the columns of P to be M2, MC, and C, QD can be found without trial and error.
- #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..
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- #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)
FAQ: Why are the columns of Q linearly independent?
What does it mean for a set of vectors to be linearly independent?
Linear independence means that no vector in the set can be written as a linear combination of the other vectors in the set. In other words, no vector in the set is redundant and each vector adds new information to the set.
How can I determine if a set of vectors is linearly independent?
A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set.
Can a set of linearly dependent vectors be transformed into a set of linearly independent vectors?
Yes, a set of linearly dependent vectors can be transformed into a set of linearly independent vectors by removing any redundant vectors and adding new vectors that are not linear combinations of the remaining vectors.
How are linear independence and matrix invertibility related?
A matrix is invertible if and only if its columns (or rows) are linearly independent. This means that if a matrix is not invertible, its columns are linearly dependent and vice versa.
Can a set of linearly independent vectors be linearly dependent when multiplied by a matrix?
Yes, a set of linearly independent vectors can become linearly dependent when multiplied by a matrix. This can happen if the matrix is singular (not invertible) or if the matrix transforms the vectors in a way that makes them linearly dependent.