Linear Algebra: Rank(A) = Rank(transpose of A)
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12:48
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Actually I think row rank and column rank are formal names for those things. "Rank" refers to the dimension of the column space, which I'm not sure what's the name of the dimension of the row space.
dalcde 1 year ago
He is showing that the row rank is equal to the column rank.
dalcde 1 year ago
But he is supposing that col rank is equal to row rank, and that is not obvious. Right?
Esaji89 1 year ago
its remarkable, you can say more in 10 minutes than professors at universities(i.e. MIT) can say in an hour!
S1CKDRIFT3R 1 year ago
apologies, i meant
minrank(Null(A))=60, minrank(Null(A^T))=0 and rank(Null(A^T))=2.
alkalait 2 years ago
ex: A is rectangular 100-by-40. 100 row-vectors can span 40-dimensional space at max. (maxrank(C(A^T))=40). AT LEAST 60 row-vectors are linearly dependent and do NOT have pivots in rref(A) (minrank(Null(A^T))=60).
40 column-vectors in 100 dimensions span at MAX 40-dimensional subspace as well (maxrank(C(A))=40,minrank(Null(A))=0). suppose it turns out rank(C(A))=38 < maxrank(C(A))=40, implies rank(Null(A))=2. implies 2 columns with no pivots in rref(A), which means 2 extra rows with no pivots.
alkalait 2 years ago
thanks--somewhat unclear in that pivot columns will have a single nonzero pivot entry=1, but pivot rows will have additional non zero entries. this should not effect the linear independence of pivot rows, but i wish you had touched on this--of course i may have it wrong
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