9) The Rank-Nullity Theorem for Linear Transformations and Matrices

Question 1

What is the Rank-Nullity Theorem

Answer 1

Question 2

What is row and column space

Answer 2

• The row space of A is the subspace of K^n spanned by the rows of A, denoted by Row(A)
• The column space of A is the subspace of K^m spanned by the columns of A, denoted by Col(A)

Question 3

What does it mean to be row or column equivalent

Answer 3

If A, B ∈ Mmn(K) are row equivalent, then Row(B) = Row(A). If A, B are column equivalent, then Col(B) = Col(A)

Question 4

What is the dimension of Row(A)

Answer 4

• The dimension of Row(A) is the number of non-zero rows in the reduced row echelon form of A
• The non-zero rows of the RREF of A form a basis for Row(A)

Question 5

What is the null space of A

Answer 5

Question 6

What is the set Null(A) a subspace of

Answer 6

K^n

Question 7

If TA : K^n → K^m is the linear transformation given by TA(v) = Av, what is Col(A)

Answer 7

Col(A) = Im(TA)
There rank(A) = rank(TA)

Question 8

What is the Rank-Nullity Theorem for matrices

Answer 8

If A is an m × n matrix
rank(A) + nullity(A) = n

Question 9

What do we know about the rank of a sqaure matrix

Answer 9