### Rank and Nullity

##### Properties of rank and nullity

Theroem 1 rankA = rankAT.

Proof The equality can be derived by comparing the two methods for computing a basis of the column space.

rankA
= number of pivots in the column echelon form of A (second method of computing a basis of colA)
= number of pivots in the row echelon form of AT (column op on A = row op on AT)
= rankAT. (first method of computing a basis of colAT)

Theorem 2 For an m by n matrix A, rankAm and n. Moreover

• rankA = m ⇔ Columns of A span RmAx = b has solutions for all b.
• rankA = n ⇔ Columns of A are linearly independent ⇔ Solution of Ax = b is unique.

Proof First,

rankA = number of pivot columns of A ≤ number of all columns of A.

Then,

rankA = rankAT ≤ number of all columns of AT = number of all rows of A.

Moreover,

rankA = m
dim(colA) = dimRm (definition of rank)
colA = Rm (take H = colA and V = Rm in this property)
⇔ The column vectors span Rm.

rankA = n
dim(nulA) = nullityA = n - rankA = 0 (equality: rank + nullity = number of columns)
nulA ={0}
Ax = 0 has only the trivial solution (definition of nulA)
⇔ The column vectors are linearly independent.