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Linear equations

Definition: A linear equation in unknowns x1, x2,—————xn is an equation that can be put in the standard form
a1 x1+ a2 x2+ —————+ an xn = b
where a1, a2,—————an and b are constants. The constant ak is called the coefficient of xk, and b is called the constant term of the equation.
System of linear equations: A system of linear equations is a list of linear equations with the same unknowns. In particular, a system of m linear equations L1, L2,…..,Lm in n unknowns x1, x2,—————xn can be put in the standard form

Where the aij and bi are constants. The number aij is the coefficient of the unknown xj in the equation Li and the number bi is the constant of the equation Li.

The system (1) is called an m×n system. It is called a square system if m=n, that is, if the number m of equations is equal to the number n of unknowns.

Homogeneous and non-homogeneous:
The system (1) is called homogeneous if all the constant terms are zero, that is, if b1=0, b2=0, b3=0, ……….., bm=0. Otherwise the system is said to be non-homogeneous.
Linear equation in unknown:
Consider the linear equation ax=b
(i) If a≠0, then x=b/a is a unique solution of ax=b.
(ii) If a=0, but b≠0, then ax=b has no solution.
(iii) If a=0 and b=0, then every scalar k is a solution of ax=b.
System of two linear equations in two unknowns (2×2) system:
Consider a system of two non-degenerate linear equations in two unknowns x and y, which can be put in the standard form
A1x+B1y=C1
A2x+B2y=C2
Since the equations are non- degenerate, A1, and B1are note both zero, and A2 and B2 are not both zero.
1) The system has exactly one solution: When the lines have distinct slopes or, equivalently, when the coefficients of x and y are not proportional.

2) The system has no solution: When the lines have the same slopes but different y intercepts, or when

3) The system has an infinite number of solutions:
When the lines have the same slopes and same y intercepts, or when the coefficients and constants are proportional,

Matrix equation of a square system of linear equation:

A system AX=B of linear equations is square if and only if the matrix A of coefficients is square.

Example: The following system of linear equations,

x+2y+3z =1

5x-6y+3z =1

x-2y+6z =1

whose coefficients matrix A

Prove that, every basis of a vector space V has the same number of elements.

Problem: Prove that, every basis of a vector space V has the same number of ...