Download Linear Algebra Lectures ::: MTL101

:::Linear Algebra Lectures ::: MTL101 :::
July-November 2015
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Lecture 1 (System of Linear Equations)
We study system of “linear equations” with complex coefficients. But for most of the cases
the coefficients will be rational numbers. Give examples of such systems. Explain what is a
solution of a system of linear equations. Geometrically describe solution of an equation when
number of unknowns is two or three and the coefficients are real numbers (respectively as a
line in a plane and a plane in space). Also describe what is a solution of a system of linear
equations when the number of unknowns is two or three (as intersection of lines or planes etc.).
Examples of systems which have (a) no solutions, (b) has a unique solution, (c) infinitely many
solutions (when number of unknowns is two or three or four):
a) x1 + x2 + x3 = 3, x1 + 2x2 + 3x3 = 6, x2 + 2x3 = 1;
b) x1 + x2 + x3 = 3, x1 + 2x2 + 3x3 = 6, x1 + x2 + 2x3 = 4;
c) x1 + x2 + x3 = 3, x1 + 2x2 + 3x3 = 6, x2 + 2x3 = 3. Verify that the first system has no
solution, (1, 1, 1) is the only solution of the second system and the third equation has infinitely
many solutions. Indeed, for any real number λ, the tuple (λ, 3 − 2λ, λ) is a solution.
Students should recall Crammer’s rule and any other methods they learnt to solve a system
of equations. The Crammer’s rule is applicable when number of unknowns and equations are
same and the determinant of the coefficient matrix is nonzero. Discuss the limitations. We will
not disucss Crammer’s rule in the class.
We write a general system of m linear equations with real (or complex) coefficients with n
unknowns x1 , x2 , . . . , xn as
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b2
·····················
···
am1 x1 + am2 x2 + · · · + amn xn = bm
ai,j (1 ≤ i ≤ m, 1 ≤ j ≤ n) are called coefficients and b1 , b2 , . . . , bm are called constant
terms of the equations. We call the system a homogenous system if bi = 0 for each i and a
nonhomogeneous system if it is not homogenous.
(5) The
by the following matrix equation: AX = Bwhere
 above system
 can bedescribed

 A=
a11 a12 a1n
b1
x1
 a21 a22 a2n 
 
 

, B =  b2  and X is the matrix (or column) of unknowns  x2 . A is
· · · · · · · · · 
· · ·
. . . 
am1 am2 amn
bm
xn
called the coefficient matrix and B the matrix (or column) of constants. The matrix (A|B) is
called the augmented



 matrix
 of
the system.
 
1 1 1 | 3
1 1 1
3
x1
In a), A = 1 2 3 , B = 6, X = x2  and (A|B) = 1 2 3 | 6 .
x
0 1 2 | 1
0 1 2
1




 
 3
1 1 1 | 3
1 1 1
3
x1
In b), A = 1 2 3 , B = 6, X = x2  and(A|B) = 1 2 3 | 6 .
x
1 1 2 | 4
1 1 2
4


 
 3


1 1 1
3
x1
1 1 1 | 3
In c), A = 1 2 3 , B = 6, X = x2  and (A|B) = 1 2 3 | 6 .
0 1 2
3
x3
0 1 2 | 3
Remark: i) A solution of a system of equations in n unknowns is an n-tuple whose components
are from R (or C or Q).
ii) The homogeneous equations always has a solution (namely, the zero tuple).
iii) When scalars are complex numbers geometric description of the solution set is not what we
have when scalars are real.
Warning: Before we write the matrix equation we should arrange the unknowns in the same
order in the equations. If an unkown is missing from an equation the corresponding coefficient
is assumed to be zero.