Presented by Rashmi Tr

Presented by Rashmi Tr. Subject Expert: Bijumon Content Editor : Madhur sir

26.5.2012

Algebra 21

Method to Find Solution of Homogeneous System of Linear Equations

Objectives From this unit a learner is expected to achieve the following 1. Get a quick review of solution of homogeneous system of linear equations. 2. Study the method to find the solution of homogeneous system of linear equations. 3. Understand the method of finding complete solution of a consistent nonhomogeneous system of linear equations. 4. Learn the definition of null space and nullity of matrix. 5. Familiarize the methods discussed in the session through examples. Sections 1. Introduction 2. Method to Find Solution of Homogeneous System of Linear Equations 3. Complete Solution of a Consistent Nonhomogeneous System of Equations. 4. Null Space and Nullity of a Matrix 5. Range of a Matrix 1. Introduction

In the previous session we have discussed the theory related to solution of homogeneous system of linear equations. In this session we describe the method of finding the solution of a homogeneous system of linear equations and illustrate it with examples. Our method of presentation is in such a way that we give a brief description of the theory

1 (already developed in the previous session) to support the method of finding the solution. We also discuss complete solution of a consistent nonhomogeneous system of linear equations. The definition of null space and nullity of a matrix will be given.

2. Method to Find Solution of Homogeneous System of Linear Equations Consider the following homogeneous system of equations:

a11 x 1+ a 12 x 2 +⋯ + a 1n x n = 0

a21 x 1+ a 22 x 2 +⋯ + a 2n x n = 0 . . . (1) ⋯ ⋯ ⋯ ⋯ ⋯

am1 x 1+ a m 2 x 2 +⋯ + a mn x n = 0

There are m linear equations in n unknowns x1, x2, . . . , xn . The system of equations (1) can be expressed as the single matrix equation AX= O . . . (2) where 0  a a ... a   x1  轾 11 12 1n 犏   x  0 a a ... a  ,  2  and 犏 A   21 22 2n  X   .  O = 犏......    .  犏. a a ... a    犏  m1 m2 mn  xn  臌犏0

A solution of the system (1) is any set of values of x1, x 2 , ... , xn which satisfy simultaneously the m equations. Equivalently, any column matrix X that satisfies the matrix equation (2) is a solution of the system (2). When the system has a solution, it is said to be consistent; otherwise, the system is said to be inconsistent. A consistent system has either a unique solution or infinitely many solutions.

The following result is required in our discussion proof of which has already been discussed in the previous session. Theorem 1 The number of linearly independent solutions of a consistent system of m linear equations in n unknowns is (n- r ), where r is the rank of the coefficient matrix A. Remarks  Thoerem 1 helps us to determine complete solution (i.e., the set of all possible solutions) to the system.  It is clear that

2 轾x1 轾0 犏x 犏0 X =犏2 = 犏犏⋮ 犏⋮ 犏犏臌xn 臌0

is always a solution to the system (2). Equivalently x1= x 2 =⋯ = xn = 0 is a set of solution to system (1). This solution is called trivial solution to the system.  In view of Theorem 1, if the rank of the matrix is equal to the number of unknowns, then the system has only the trivial solution.  If rank is less than the number of unknowns, then the system has non-trivial solutions. We assign arbitrary values to (n  r) unknowns and determine the remaining r unknowns uniquely in terms of these (n  r) values, where n is the number of unknowns and r is rank of .the coefficient matrix

Method To find the solution of a homogenous system of m equations in n unknowns, we proceed as follows: 1. Write down the corresponding matrix equation AX = O and find the rank r of the matrix A. Case 1 If r = n, i.e., if rank of A = number of unknowns, then the system has only the trivial solution. Case 2 If r < n i.e., if rank of A < number of unknowns, then the system has infinitely many solutions. We assign arbitrary values to (n  r) unknowns and determine the remaining r unknowns uniquely in terms of these (n  r) values.

Example 1 Solve completely the system of equations x + 3 y – 2 z = 0 2 x  y + 4 z = 0 x – 11 y + 14 z = 0

3 Solution The matrix equation corresponding to the given system of equations is given by

1 3  2 x 0       2 1 4 y  0 …(3) 1 11 14 z 0 Obviuously,

轾x 轾0 犏犏 X=犏 y = 犏0 臌犏z 臌犏0 is a solution (called the trivial solution) to the system. Now we find non-trivial solutions, if they exist. Now to determine the rank, we reduce A to its echelon form as follows:

1 3  2   A  2 1 4 1 11 14

轾1 3- 2 ~犏 0- 7 8 byR ( - 2) 犏 21 臌犏0- 14 16R31 ( - 1)

轾1 3- 2 ~犏 0 1-8 byR ( - 1 ) 犏 72 7 臌犏0- 14 16

10 轾1 0 7 ~犏 0 1-8 byR ( - 3) 犏 7 12 犏臌0 0 0R32 (14) The matrix given just above is the echelon form of A. Since the echelon form has 2 non- zero rows, rank of A is 2. Since rank of A = 2 < 3 = the number of unknowns, the system possesses non-trivial solutions. The given matrix equation (3) is equivalent to the matrix equation

10 1 0 7  x 0   0 1  8 y  0  7      . . . (4)   0 0 0 z 0

4 Now the matrix equation (4) is equivalent to the system of equations

10 x  7 z  0 8 y  7 z  0

We can assign arbitrary values to n  r = 3  2 = 1 unknown, say to the variable z. Take z = a, where a is arbitrary real number. Then we get from the first equation

10 10 x  7 a  0 or x   7 a and from the second equation

8 8 y  7 a  0 or y  7 a . Hence the solution is

 10  x  7 a     X  y   8 a .   7   z    a  or equivalently

10 8 x= -7 a, y = 7 a , z = a , where a is arbitrary real number. Example 2 Does the following system of equations possess a non-trivial solution? x  2y  3z  0 2x  3y  z  0 4x  y  2z  0

Solution

轾1 2- 3 A =犏2 - 3 1 Here 犏

臌犏4- 1 - 2

轾1 2- 3 ~犏 0- 7 7 byR ( - 2) 犏 21 臌犏0- 9 10R31 ( - 4)

轾1 2- 3 ~犏 0 1- 1 byR ( - 1 ) 犏 2 7 臌犏0- 9 10

轾1 0- 1 ~犏 0 1- 1 byR ( - 2) 犏 12 臌犏0 0 1R32 (9)

5 轾1 0 0 ~犏 0 1 0 byR (1) 犏 13 臌犏0 0 1R23 (1)

1 0 0   Hence the row echelon matrix of A is 0 1 0 and since it has 3 non-zero rows, rank of 0 0 1 A is 3. Since rank is the same as the number of unknowns, the system has only one solution and it is the trivial solution x  0, y  0, z  0. Therefore, the given system has no non-trivial solution.

3. Complete Solution of a Consistent Non-homogeneous System of Equations The solution of system of nonhomogeneous linear equations will be discussed in detail in the next session. However, now, we note that all the solutions of a system of nonhomogeneous linear equations can be determined if we know complete solution of the corresponding homogeneous system of equations and a particular solution of the nonhomogeneous system of equations. That is, if the system of non-homogeneous equations or its equivalent matrix equation AX= B is consistent, a complete solution of the non-homogenous system is given by

X= XH + X P , where X H is the complete solution of the homogeneous system AX = 0 and X P is any particular solution of the nonhomogeneous system AX= B . Example 3 Solve the following system

x1-2 x 2 + 3 x 3 = 4 x1+ x 2 +2 x 3 = 5

Solution

Setting x1 = 0; we have x3=2 and x 2 = 1. Hence a particular solution of the non-homogenous system is

轾0 X P = 犏1 . 臌犏2 It can be seen that the complete solution of the corresponding homogeneous system

x1-2 x 2 + 3 x 3 = 0 x1+ x 2 +2 x 3 = 0

6 轾-7a is XH = 犏 a , where a is an arbitrary real number. Hence the complete solution of the 臌犏3a given non-homogeneous system is

轾-7a 轾 0 轾 - 7 a X= XH + X P =犏 a + 犏1 = 犏 1 + a . 臌犏3a 臌犏 2 臌犏 2+ 3 a

4. Null Space and Nullity of a Matrix The proof of the following result has been discussed in the previous session. Theorem 2 The set of solutions of the homogeneous matrix system of linear equations(1) is a subspace of the vector space V ( ) of column matrices with elements from the field n n 1 of real numbers.

Definition The subspace, mentioned in the above theorem, generated by the column vectors X such that AX= O is called the column null space of the m n matrix A and its dimension n- r ,( where r is the rank of the matrix,) is called the column nullity of the matrix. In view of the Definition, we have rank + column nullity = number of columns.

Definition. The subspace generated by the row matrices Y such that YA= O is called the row null space and its dimension m- r is called the row nullity of the matrix. Also, we note that rank + row nullity = number of rows. We note that for square matrices, number of columns and number of rows are equal. Hence the above observations yield the following result: Theorem 3 The row nullity and the column nullity of the square matrix are equal. In view of theorem we simply say nullity of square matrices, where it is both row nullity and column nullity. Hence for an n-square matrix,

7 In view of above theorem , as the row nullity and column nullity are the same, we simply say nullity of a square matrix for this common value, Hence for an n-square matrix rank + nullity = n.

Theorem 4 If A, B are two n-rowed square matrices, then max{v ( A )祝 v ( B )} v ( AB )� v ( A ) v ( B ) where v( A ) , v( B ) v( AB ) denote the nullities of the square matrices A, B and AB respectively. Proof. We recall the inequality

r(A )+ r ( B ) - n＃ r ( AB ) min{ r ( A ), r ( B )} …(5)

Now r(A )= n - v ( A ), r ( B ) = n - v ( B ), r ( AB ) = n - v ( AB ). Substituting these values in (5) we obtain the result. 5. Range of a Matrix

Let A be an m n matrix. If X is any column vector of the vector space Vn ( ) , then AX is a column vector of the vector space Vm ( ) . Definition Let A be an m n matrix. When the column vector X ranges over the entire space V ( ) , then the vector AX ranges over a subset of the space V ( ) . This subset is n m called the range of the matrix A. Symbolically,

range of A = {AX挝 Vm ( ) | X V n ( )}.

Theorem 5 The range of an m n matrix A is a subspace of Vm ( ) . Proof

Let Y1, Y 2 range of A . i.e., Y1, Y 2 挝 { AX | X Vn ( )}. Then

Y1= AX 1, Y 2 = AX 2

for some X1, X 2 Vn (F ). Now

cY1+ Y 2 = cAX 1 + AX 2 = A( cX 1 + X 2 ).

8 Hence cY1+ Y 2 range ofA . This shows that the range of an m n matrix A is a subspace of

Vm (F ) .

range of A is also called range space of A.

Column Space The subspace generated by the column vectors of A is called the column space of A. i.e, column space of A is the set of linear combinations of columns of A. Summary In this session we have described the method of finding the solution of a homogeneous system of linear equations. We have also discussed complete solution of a consistent nonhomogeneous system of linear equations. The definition of null space and nullity of matrix, and the statement of Sylvester’s Law of Nullity has been given. Hope you enjoy this session. See you next time. Now try to attempt the following assignments.

Assignments Find all the non-trivial solutions of the following system of equations. 1. 7 x + y  2 z = 0 ; x + 5 y  4 z = 0 ; 3 x  2 y + z = 0. 2. x  y + z = 0 ; 2 x + y  z = 0 ; x + 5 y  5 z = 0. 3. Determine the values of a, if any, for which the system of equations a x  2 y + z = 0 a x + (1  a) y + z = 0

2 x  y + 2a z =0 has a non-trivial solution and find such solutions when they exist. 4. Solve completely the following system:

4x1- x 2 + 2 x 3 + x 4 = 0 2x+ 3 x - x - 2 x = 0 1 2 3 4 7x2- 4 x 3 - 5 x 4 = 0 2x1- 11 x 2 + 7 x 3 + 8 x 4 = 0 Quiz

1. If an n n matrix A is reduced to echelon form C and C has a zero row, then the homogenous system AX = 0 has a ______. (a) nontrivial solution (b) only the trivial solution

9 (c) only one nontrivial solution (d) none of the above Ans. (a) 2. If A is an n n matrix, then homogeneous system of linear system AX = 0 has a nontrivial solution if and only if ______(a) its determinant not equal to 0 (b) its determinant = 0 (c) its determinant = 1 (d) none of the above Ans. (b) 3. Consider the m × n homogeneous linear system AX = 0 , where M = [A 0] is the augmented matrix then ______(a) rank (A) < rank (M) (b) rank (A) > rank (M) (c) rank (A) = rank (M) (d) none of the above Ans. (c)

FAQ 1. What do you mean by a subspace? Ans. Let V be a vector space over the field R of real numbers. A non empty subset W of V is a subspace of V if W itself is a vector space over R with the operations of vector addition and scalar multiplication already defined in V. 2. What do you mean by basis?

Ans. Let V be a vector space. A subset B of V is said to be a basis for V if (i) B spans V and (ii) B is linearly independent in V.

Glossary

10 System of linear equations: A system of m linear equations in n unknowns x1, x2, . .

’ ’ . , xn is a set of equations of the following form, where the coefficients a j k s and b j s are real numbers:

a11 x 1+ a 12 x 2 +⋯ + a 1n x n = b 1

a21 x 1+ a 22 x 2 +⋯ + a 2n x n = b 2 ⋯ ⋯ ⋯ ⋯ ⋯ . . . (1)

am1 x 1+ a m 2 x 2 +⋯ + a m n x n = b m

Homogeneous and Non-Homogeneous System of linear equations: The system of equations (1) is said to be homogeneous if all the bj are zero; otherwise, it is said to be non-homogeneous. Solution of System of Linear Equations: A solution of the system (1) is any set of values of x1, x 2 , ... , xn which satisfy simultaneously the m equations. Consistent and Inconsistent System of Linear Equations: When the system has a solution, it is said to be consistent; otherwise, the system is said to be inconsistent. A consistent system has either a unique solution or infinitely many solutions. Solution to the matrix equation: A column matrix X is a solution to the matrix equation AX = B if it satisfies the matrix equation.

轾0 犏0 犏 Trivial solution: The solution X= O = 犏. to the homogeneous system of linear 犏. 犏臌犏0 equations is called the trivial solution. Column null space and column nullity of a matrix: The subspace generated by the column vectors X such that AX= O is called the column null space of the m n matrix A and its dimension n- r , where r is the rank of the matrix, is called the column nullity of the matrix. Row null space and row nullity of a matrix: The subspace generated by the row matrices Y such that YA= O is called the row null space and its dimension m- r is called the row nullity of the matrix.

11 Range of a Matrix or range space of A: Let A be an m n matrix. When the column vector X ranges over the entire space Vn ( ) , then the vector AX ranges over a subset of the space . This subset is called the range of the matrix A. Symbolically, Vm ( )

range of A = {AX挝 Vm ( ) | X V n ( )}.

REFERENCES Books 1. I.N. Herstein, Topics in Algebra, Wiley Easten Ltd., New Delhi, 1975. 2. K. B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt. Ltd., New Delhi, 2000. 3. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, First Course in Linear Algebra, Wiley Eastern, New Delhi, 1983. 4. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra (2nd Edition), Cambridge University Press, Indian Edition, 1997., New Delhi, 1983. 5. S.K. Jain, A. Gunawardena and P.B. Bhattacharya, Basic Linear Algebra with MATLAB, Key College Publishing (Springer-Verlag), 2001.

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