Rank Nullity Theorem

Question 1

What is nullity of linear map

Answer 1

Nullity of a linear map is dim(ker(map))

Question 2

What is nullity of matrix A

Answer 2

Nullity of matrix A is dim(Na) (null space of A)

Question 3

What is null space of A

Answer 3

Nullspace of A is solution space of Ax =0 number of parameters in general solution to Ax =0 number of non-pivot variables = see daniels help = variable multiplied by a non-pivot

Question 4

What does the rank nullity theorem state

Answer 4

The rank nullity theorem states that:
Rank(phi)+ null(phi) = dim V
Where phi is map V to W
#pivot variables + #non-pivt variables = # variables = n

Question 5

When does map phi have maximal rank

Answer 5

Map phi has maximal rank when rank(phi) = min(dim V, dim W)

Question 6

What is rank of a matrix A equal to

Answer 6

Rank of a matrix A is equal to:
Column rank of A
Row rank of A
Smallest r such that B*C =A where B is m,r and C is r,n

Question 7

What is column rank of matrix A

Answer 7

Column rank of matrix A is:

Dim(cols span(A))

Question 8

What is row rank of matrix A

Answer 8

Row rank of matrix A is:

Dim(rows span(A)) = no. L.I rows