Rank Flashcards
Rank in term of independent vectors
R(A) = number of independent vectors of A
Relation between rank and determinant
If |A| = 0 then rank < n
If |A| ≠ 0 then rank = n
Rank of matrix in terms of minor
Rank(A) = highest order of non zero minor of A
Which matrix has rank = 0
Only null matrix have rank zero
Effect of elementary transformation on rank
No effect
Rank(Aᵀ)
Rank(Aᵀ) = Rank(A)
Rank of AAᵀ
Rank of AAᵀ = Rank of AᵀA = Rank of A
Rank(AB)
Rank(AB) ≤ min(Rank A, Rank B)
Rank (A+B)
Rank (A+B) ≤ Rank(A) + Rank(B)
Rank of A-B
Rank (A-B) ≥ Rank A - Rank B
Rank of Amxn
Rank A ≤ min{m,n}
Rank of Iₙ
Rank of Iₙ = n
Rank of Oₙₓₙ
Rank of Oₙₓₙ = 0
Rank of diagonal matrix
non zero diagonal elements
Vector space of polynomial of degree <= n
Also dimension of it
Basis of matrix M of 2x3 order
Explain dimensions of W be a real subspace of the real space R³
Explain relation linear dependency or independency if you have one real number and another one as complex number
If you have one real number and one complex number (where the complex has non-zero part), then they are linearly independent. This is because the only solution to linear combination of these two numbers being zero is if both coefficients in the combination are zero.
