In this presentation you can learn about row,column,null space and also learn about rank & nullity

1. GANDHINAGAR INSTITUTE OF
TECHNOLOGY
MECHANICAL DEPARTMENT
VECTOR CALCULUS AND LINEAR ALGEBRA

2. NAME :- PARTHIV PAL
ENROLLMENT NO. :- 160120119062
SEMESTER :- 2
TOPIC :- ROW SPACE,COLUMN SPACE AND NULL SPACE & RANK
AND NULLITY
GUIDED BY :- PROF SANDIP PANCHAL

3. • ROW SPACE:-
The subspace of Rn spanned by the row vectors of A is called the
row space of A.
• COLUMN SPACE:-
The subspace of Rm spanned by the column vector of A is called
the column space of A.
• NULL SPACE:-
The solution space of the homogeneous system of equations
AX=0 is called the null space of A.

4. BASIS FOR ROW SPACE:-
• THEOREM 1:- Elementary row transformation do not change the row
space and null space of the matrix.
• THEOREM 2:- If B is the row echelon form of A then the row vectors
of B with leading 1’s form of the basis for the row space of B, hence
form a basis for the row space of A.

5. BASIS FOR COLUMN SPACE :-
THEOREM 1 :- If A and B are the row equivalent matrices then,
(1) A set of the column vector of matrix A is linearly independent
if and only if the corresponding column vector of B are linearly
independent.
THEOREM 2 :- If B is the row echelon form of the matrix then,
(1) The column vector containing the leading 1’s of the row
vector from a basis for the column space of B

6. NULL SPACE:-
• The basis for the null space of A is the basis for the solution space of
the homogeneous system AX=0.

7. Example:- Find a basis for the row and column spaces of A.
solution R21-2R1 , R3+R1
R3+R2

8. (-1/7)R2











0000
7/4110
2541
A =B
Basis for the row space of A = Non zero row of B = {(1,4,5,2) , (0,1,1,4/7)}
Leading 1’s appear in column 1 and 2.
BasisforthecolumnspaceofA= { 









3
1
4
}

9. RANK AND NULLITY :-
• The dimension of row/column space of a matrix A(the no. of the non
zero rows in the row echelon form of A) is called the rank of A and is
denoted by
• If A is an m*n matrix then,
rank(A) < min (m,n)
Thus the largest possible value of rank(A)=min(m,n) where min(m,n)
means the smaller of m and n.
Ρ(A)

10. NULLITY :-
• The dimension of the null space of a matrix A is called the nullity of A
and is denoted by nullity(A).
DIMENSION THEOREM :-
• If A is an m*n matrix then,
rank(A)+nullity(A)=n(number of columns)
• If A is an m*n matrix then nullity(A) represents the number of
parameter in the general solution of AX=0.

11. Example:-Find the rank and nullity of the matrix.
Solution:- Reducing matrix to the row echelon form,
(1/2)R1













000
204
2/101
A

12. R2-4R1









 

000
000
2/101
A =B
Rank(A) = Number of non zero rows =1
Nullity (A) = n – rank(A)
= 3 – 1
= 2

13. THANK YOU…