**Let us look at some examples of linear systems:**

**Suppose a, b ∈ Consider the system ax = b.**

**If a 6= 0 then the system has a unique solution**

**If a = 0 and:**

** a) b 6= 0 then the system has no solution.**

** b) b = 0 then the system has infinite number of solutions, namely all x ∈ R.**

**2. We now consider a system with 2 equations in 2 unknowns.**

** Consider the equation ax + by = c. If one of the coefficients, a or b is non-zero, then this linear equation represents a line in R2. Thus for the system**

** **

**the set of solutions is given by the points of intersection of the two lines. There are three cases to be considered. Each case is illustrated by an example.**

** a) Unique Solution**

**Observe that in this case, a**_{1}b_{2}− a_{2}b_{1}6= 0.

**b) Infinite Number of Solutions**

**x + 2y = 1 and 2x + 4y = 2. The set of solutions is**

**(x, y)**^{t}= (1 − 2y, y)^{t}= (1, 0)^{t}+ y(−2, 1)^{t}

**with y arbitrary. In other words, both the equations represent the same line.**

**Observe that in this case, a**_{1}b_{2}− a_{2}b_{1}= 0, a_{1}c_{2}− a_{2}c_{1}= 0 and b_{1}c_{2}− b_{2}c_{1}= 0.

**c) No Solution**

**x + 2y = 1 and 2x + 4y = 3. The equations represent a pair of parallel lines and hence there is no point of intersection.**

**Observe that in this case, a1b2 − a**_{2}b_{1}= 0 but a_{1}c_{2}− a_{2}c_{1}6= 0.

**3.**

**As a last example, consider 3 equations in 3 unknowns.**

**A linear equation ax + by + cz = d represent a plane in R3 provided (a, b, c) 6= (0, 0, 0). As in the case of 2 equations in 2 unknowns, we have to look at the points of intersection of the given three planes. Here again, we have three cases. The three cases are illustrated by examples.**

** a) Unique Solution**

** Consider the system x+y+z = 3, x+4y+2z = 7 and 4x+10y−z = 13. The unique solution to this system is (x, y, z) ^{t} = (1, 1, 1)^{t}; i.e. the three planes intersect at a point.**

**b) Infinite Number of Solutions**

**Consider the system x + y + z = 3, x + 2y + 2z = 5 and 3x + 4y + 4z = 11. The set of solutions to this system is (x, y, z)**

^{t}= (1, 2− z, z)^{t }= (1, 2, 0)^{t}+ z(0,−1, 1)^{t}, with z arbitrary:**the three planes intersect on a line.**

**c) No Solution**

**The system x + y + z = 3, x + 2y + 2z = 5 and 3x + 4y + 4z = 13 has no solution. In this case, we get three parallel lines as intersections of the above planes taken two at a time.**

**Definition and a Solution Method**

**Linear System: A linear system of m equations in n unknowns x _{1}, x_{2}, . . . , x_{n} is a set of equations of the form**

**a _{11}x_{1} + a_{12}x_{2} + · · · + a_{1n}x_{n} = b_{1}**

**a _{21}x_{1} + a_{22}x_{2} + · · · + a_{2n}x_{n} = b_{2}**

**a _{m1}x_{1} + a_{m2}x_{2} + · · · + a_{mn}x_{n} = b_{m}**

**where for 1 ≤ i ≤ n, and 1 ≤ j ≤ m; a _{ij} , b_{i} ∈ R. Linear System (2.2.1) is called homogeneous if b_{1} = 0 = b_{2} = · · · = b_{m} and non-homogeneous otherwise.**

**We rewrite the above equations in the form Ax = b, where**

**The matrix A is called the coefficient matrix and the block matrix [A b] , is the augmented matrix of the linear system.**

**Remark: Observe that the i ^{th} row of the augmented matrix [A b] represents the i^{th} equation and the j^{th} column of the coefficient matrix A corresponds to coefficients of the j^{th} variable x_{j} . That is, for 1 ≤ i ≤ m and 1 ≤ j ≤ n, the entry a_{ij} of the coefficient matrix A corresponds to the i^{th} equation and j^{th} variable x_{j} ..**

**For a system of linear equations Ax = b, the system Ax = 0 is called the associated homogeneous system.**

**Solution of a Linear System: A solution of the linear system Ax = b is a column vector y with entries y1, y2, . . . , yn such that the linear system.**

**That is, if yt = [y1, y2, . . . , yn] then Ay = b holds.**

**Note: The zero n-tuple x = 0 is always a solution of the system Ax = 0, and is called the trivial solution. A non-zero n-tuple x, if it satisfies Ax = 0, is called a non-trivial solution.**

**Example1 Let us solve the linear system x + 7y + 3z = 11, x + y + z = 3, and**

**4x + 10y − z = 13.**

**The above linear system and the linear system**

**Interchange the first two equations.****Eliminating x from 2**^{nd}and 3^{rd}equation, we get the linear system

**(obtained by subtracting the first equation from the second equation.)**

**(obtained by subtracting 4 times the first equation from the third equation.)****Eliminating y from the last two equations of system, we get the system**

**obtained by subtracting the third equation from the second equation.****Now, z = 1 implies**

**and x = 3−(1+1) = 1. Or in terms of a vector, the set of solution is { (x, y, z)**^{t}: (x, y, z) = (1, 1, 1)}.