Temperament addition: Difference between revisions
Cmloegcmluin (talk | contribs) make corrections re: linear (in)dependence vs. collinearity, as well as vectors vs. basis vectors (thanks for the feedback, Sintel!) |
Cmloegcmluin (talk | contribs) →Sintel's proof of the linear independence conjecture: make it readable at least, oops |
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If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (i'll use A and B as their rowspace equivalently): | If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (i'll use A and B as their rowspace equivalently): | ||
dim(A + B) - m = dim(ker(A) + ker(B)) - (n-m) | dim(A + B) - m = dim(ker(A) + ker(B)) - (n-m) | ||
>> dim(A)+dim(B)=dim(A+B)+dim(A∩B) => dim(A + B) = dim(A) + dim(B) - dim(A∩B) | >> dim(A)+dim(B)=dim(A+B)+dim(A∩B) => dim(A + B) = dim(A) + dim(B) - dim(A∩B) | ||
dim(A) + dim(B) - dim(A∩B) - m = dim(ker(A) + ker(B)) - (n-m) | dim(A) + dim(B) - dim(A∩B) - m = dim(ker(A) + ker(B)) - (n-m) | ||
>> by duality of kernel, dim(ker(A) + ker(B)) = dim(ker(A ∩ B)) | |||
>> by duality of kernel, dim(ker(A) + ker(B)) = dim(ker(A ∩ B)) | |||
dim(A) + dim(B) - dim(A∩B) - m = dim(ker(A ∩ B)) - (n-m) | dim(A) + dim(B) - dim(A∩B) - m = dim(ker(A ∩ B)) - (n-m) | ||
>> rank nullity: dim(ker(A ∩ B)) + dim(A ∩ B) = n | >> rank nullity: dim(ker(A ∩ B)) + dim(A ∩ B) = n | ||
dim(A) + dim(B) - dim(A∩B) - m = n - dim(A ∩ B) - (n-m) | dim(A) + dim(B) - dim(A∩B) - m = n - dim(A ∩ B) - (n-m) | ||
m + m - dim(A∩B) - m = n - dim(A ∩ B) - (n-m) | m + m - dim(A∩B) - m = n - dim(A ∩ B) - (n-m) | ||
m + m - m = n - n + m | m + m - m = n - n + m | ||
m = m | m = m | ||