Study Notes on Rank for GATE 2018

Rank of Matrix

Let

The i’th row of A is

and the j’th column of A is

Definition of row space and column space

span{row₁(A), row₂(A),…, row_m(A)},

which is a vector space under standard matrix addition and scalar multiplication, is referred to as the row space. Similarly,

span{col₁(A), col₂(A), …, col_n(A)},

which is also a vector space under standard matrix addition and scalar multiplication, is referred to as the column space.

A matrix B is row equivalent to a matrix A if B result from A via elementary row operations.

Result

If A and B are two m×n row equivalent matrices, then the row spaces of A and B are equal.

Finding base of row and column spaces

Suppose A is a m×n matrix. Then, the bases of the row and column spaces can be found via the following steps.

Step 1:

Transform the matrix A to the matrix in reduced row echelon form.

Step 2:

• The nonzero rows of the matrix in reduced row echelon form a basis of the row space of A.
• The columns corresponding to the ones containing the leading 1’s form a basis. For example, if n=6 and the reduced row echelon matrix is

then the 1^st, 3^rd and 4^th columns contain a leading 1 and thus col₁(A), col₃(A), col₄(A) form a basis of the column space of A.