Rank

Definition, Rank(AB), Rank(A+B)

Definition and calculation of rank, row and column rank, rank(AB), rank(A+B), geometric interpretation and rank identities.

Chapter contents

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Topic	Pages
Definition & calculation of rank	1
Row rank = Col rank ⇔ rank(M) = rank(Mᵀ)	1–2
rank(AB) ≤ rank(B) & rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	2–4
Visual illustration of rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	5–6
Extension of rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	6–8
rank(AB) ≤ rank(A) & rank(AB) = rank(A) − dim( row(A) ∩ ℓ-null(B) )	8–9
Multiplication by invertible matrix	9–10
Geometric view of rank(AB)	10–11
Geometric view of rank(AB) < min (rank(A), rank(B))	11–12
rank(AᵀA) = rank(A)	12–13
rank(A+B) ≤ rank(A) + rank(B) − dim( col(A) ∩ col(B) )	13–15
Comparison between AB & A+B	15

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