Null space and rank nullity theorem

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The document outlines the null space and dimension theorem in linear algebra, defining null space and its general solution in terms of free variables. It discusses the decomposition of vectors and the relationship between rank, nullity, and the number of columns in a matrix, ultimately verifying the dimension theorem. The content also includes references to inspiration from a professor and additional resources in linear algebra.

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 Definition: The null space of an matrix A,
written as Nul A, is the set of all solutions of the
homogeneous equation In set notation,
.
m n
Nul {x : x is in and x 0}n
A A ¡
x 0A 

 Solution: The first step is to find the general solution
of in terms of free variables.
 Row reduce the augmented matrix to reduce
echelon form in order to write the basic variables in
terms of the free variables:
3 6 1 1 7
1 2 2 3 1
2 4 5 8 4
A
   
   
 
   
x 0A 
 0A

 =
 The general solution is ,
, with x2, x4, and x5 free.
 Next, decompose the vector giving the general
solution into a linear combination of vectors where the
weights are the free variables. That is,
1 2 0 1 3 0
0 0 1 2 2 0
0 0 0 0 0 0
  
 
 
  
1 2 4 5
3 4 5
2 3 0
2 2 0
0 0
x x x x
x x x
   
  

1 2 4 5
2 3x x x x  
3 4 5
2 2x x x  

 Every linear combination of u, v, and w is an element
of Null A.
1 2 4 5
2 2
3 4 5 2 4 5
4 4
5 5
2 3 2 1 3
1 0 0
2 2 0 2 2
0 1 0
0 0 1
x x x x
x x
x x x x x x
x x
x x
           
         
         
              
         
         
                 
2 4 5
u v wx x x  

 If A is an m×n matrix, then:
a) rank(A)=the number of leading variables in the
solution of Ax=0.
b) nullity(A)=the number of parameters in the
general solution of Ax=0.
 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

 Prove the dimension theorem for
 Augmented matrix is,

 Corresponding system of equations are:

 Rank (A) = 2
 Nullity (A) = 2
 Dimension (A) = 4 = no. of columns
 rank(A) + nullity (A) = 2 + 2
= 4
= no. of columns
 So, here dimension theorem is verified.

 Inspiration from Prof. Bhavesh V. Suthar
 Notes of Vector Calculus Linear Algebra
 Textbook of VCLA
 Image from Google images
 Some my own knowledge

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Null space and rank nullity theorem

1. Name: Panchal Dhrumil Indravadan Activity: VCLA(ALA) Branch: Computer Engineering(B.E.) Semester: Second Sem Year: 2017-18

2. Null space and Dimension theorem

3.  Definition: The null space of an matrix A, written as Nul A, is the set of all solutions of the homogeneous equation In set notation, . m n Nul {x : x is in and x 0}n A A ¡ x 0A 

4.  Solution: The first step is to find the general solution of in terms of free variables.  Row reduce the augmented matrix to reduce echelon form in order to write the basic variables in terms of the free variables: 3 6 1 1 7 1 2 2 3 1 2 4 5 8 4 A               x 0A   0A

5.  =  The general solution is , , with x2, x4, and x5 free.  Next, decompose the vector giving the general solution into a linear combination of vectors where the weights are the free variables. That is, 1 2 0 1 3 0 0 0 1 2 2 0 0 0 0 0 0 0           1 2 4 5 3 4 5 2 3 0 2 2 0 0 0 x x x x x x x         1 2 4 5 2 3x x x x   3 4 5 2 2x x x  

6.  Every linear combination of u, v, and w is an element of Null A. 1 2 4 5 2 2 3 4 5 2 4 5 4 4 5 5 2 3 2 1 3 1 0 0 2 2 0 2 2 0 1 0 0 0 1 x x x x x x x x x x x x x x x x                                                                                      2 4 5 u v wx x x  

7.  If A is an m×n matrix, then: a) rank(A)=the number of leading variables in the solution of Ax=0. b) nullity(A)=the number of parameters in the general solution of Ax=0.  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

8.  Prove the dimension theorem for  Augmented matrix is,   Corresponding system of equations are:

9.  Which can be expressed as follow:

10.  Rank (A) = 2  Nullity (A) = 2  Dimension (A) = 4 = no. of columns  rank(A) + nullity (A) = 2 + 2 = 4 = no. of columns  So, here dimension theorem is verified.

11.  Inspiration from Prof. Bhavesh V. Suthar  Notes of Vector Calculus Linear Algebra  Textbook of VCLA  Image from Google images  Some my own knowledge

Null space and rank nullity theorem

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