System of Linear Equations

System of Linear Equations: In mathematics, a system of linear equations, also known as a linear system, consists of one or more equations that are linear and involve the same set of variables. For instance, a linear system involving three variables corresponds to a set of planes. The solution to this system is the point at which all the equations are simultaneously satisfied, representing the intersection of these planes.

This article will explain what a system of linear equations is, how to solve them using different methods, and provide examples to help you understand.

A system of linear equations is a mathematical concept involving two or more equations that are linear in nature and share the same variables. These equations collectively define relationships between variables, often representing lines, planes, or higher-dimensional surfaces in space. The solution to the system is the set of values for the variables that satisfies all equations simultaneously, typically corresponding to points of intersection or commonality among the equations.

Let A=[a_ij] _nxn is a square matrix of order n, then the sum of diagonal elements is called the trace of a matrix which is denoted by tr(A). tr(A) = a₁₁ + a₂₂ + a₃₃+ ……….+ a_nn 

Let A and B be any two square matrices of order n, then  

1. tr(kA) = k tr(A) where k is a scalar.
2. tr(A+B) = tr(A)+tr(B)
3. tr(A-B) = tr(A)-tr(B)
4. tr(AB) = tr(BA)

Linear equations can have three kinds of possible solutions:  

• No Solution
• Unique Solution
• Infinite Solution

Rank of a matrix: The rank of the matrix is the number of non-zero rows in the row reduced form the maximum number of independent rows or the maximum number of independent columns. 
Let A be any m x n matrix and it has square sub-matrices of different orders. A matrix is said to be of rank r if it satisfies the following properties:  

1. It has at least one square sub-matrices of order r and has a non-zero determinant.
2. All the determinants of square sub-matrices of order (r+1) or higher than r are zero.

Rank is denoted as P(A). 
if A is a non-singular matrix of order n, then the rank of A = n i.e. P(A) = n. 

Properties of rank of a matrix:  

1. If A is a null matrix then P(A) = 0 i.e. Rank of null matrix is zero.
2. If I_n is the nxn unit matrix then P(A) = n.
3. Rank of a matrix A mxn , P(A) ? min(m,n). Thus P(A) ?m and P(A) ? n.
4. P(A _nxn ) = n if |A| ? 0
5. If P(A) = m and P(B)=n then P(AB) ? min(m,n).
6. If A and B are square matrices of order n then P(AB) = P(A) + P(B) – n.
7. If A_m×1 is a non zero column matrix and B_1×n is a non zero row matrix then P(AB) = 1.
8. The rank of a skew symmetric matrix cannot be equal to one.

System of homogeneous linear equations AX = 0.  

1. X = 0. is always a solution; means all the unknowns has same value as zero. (This is also called trivial solution)
2. If P(A) = number of unknowns, unique solution.
3. If P(A) < number of unknowns, infinite number of solutions.

System of non-homogeneous linear equations AX = B.  

1. If P[A:B] ?P(A), No solution.
2. If P[A:B] = P(A) = the number of unknown variables, unique solution.
3. If P[A:B] = P(A) ? number of unknown, infinite number of solutions.

Here P[A:B] is the rank of Gauss Elimination representation of AX = B. 
There are two states of the Linear equation system:  

• Consistent State: A System of equations having one or more solutions is called a consistent system of equations.
• Inconsistent State: A System of equations having no solutions is called inconsistent system of equations.

Linear dependence and Linear independence of vector: 
Linear Dependence: A set of vectors X₁ ,X₂ ….X_r is said to be linearly dependent if there exist r scalars k₁ ,k₂ …..k_r such that: k₁ X₁ + k₂X₂ +……..k_r X_r = 0. 

Linear Independence: A set of vectors X₁ ,X₂….X_r is said to be linearly independent if for all r scalars k₁,k₂ …..k_r such that k₁X₁+ k₂ X₂+……..k_rX_r = 0, then k₁ = k₂ =……. = k_r = 0. 

The subsequent techniques for solving the system of linear equations AX = B are viable solely under the condition that the coefficient matrix A is non-singular, meaning |A| ≠ 0.

These methods include:

• Cramer’s approach
• Inverse method
• Gauss-Jordan method
• Gauss Elimination method
• LU Decomposition method of factorization (also known as Method of Triangularization)

How to determine linear dependency and independency ? 
Let X₁, X₂ ….X_r be the given vectors. Construct a matrix with the given vectors as its rows.  

1. If the rank of the matrix of the given vectors is less than the number of vectors, then the vectors are linearly dependent.
2. If the rank of the matrix of the given vectors is equal to number of vectors, then the vectors are linearly independent.