Lecture 02.pdf linear algebra importantt

Linear algebra is the mathematics of structure and transformation. It studies how quantities relate linearly—meaning through addition and scalar multiplication—and how these relationships can be represented, manipulated, and understood using vectors, matrices, and linear maps.
It’s not just about solving equations—it’s about understanding the geometry of space, the behavior of systems, and the logic of transformation.

1. LINEAR ALGEBRA
BSCS (SEMESTER Ⅱ)
Lecture# 02
June 10, 2020
Section -B
Amna Tahir
GCU, Lahore

2. 2
OutlineSystem of Linear Equations
Solution of Linear Systems
Cramer’s Rule

3. System of Linear Equations
•An equation of the form
�
�??????
�+�
�??????
�+...+�
????????????
??????=�
is called a linear equation in nvariables (unknowns) �
1, �
2, . . . , �
�, where
�
1, �
2, . . . , �
�andb are constants (usually real numbers).
•A collection (list or set) of several such linear equations with the same set of
variables is called a system of linear equationsor briefly called linear
systems.
Note:The system is called linear because each variable {�
1, . . . , �
�}
appears in the first power only.
3

4. System of Linear Equations
A general system of m linear equations in n variables can be written as:
�
11�
1+�
12�
2+...+�
1��
�=�
1
�
21�
1+�
22�
2+...+�
2��
�=�
2
----------------- (1)
-----------------
�
�1�
1+�
�2�
2+...+�
���
�=�
�
where
•�
1, �
2, . . . , �
�are the variables (unknowns).
•�
11,�
12,...,�
��are the coefficients of the system.
•�
1,�
2,...,�
�are the constant terms.
4

5. System of Linear Equations
•The system (1) is called an �×�system. It is called a square system if
�=�,that is, the number of equations(m) is equal to the number of
variables(n).
Examples:
5
•2×2linearsystem
3�
1−2�
2=−1
4�
1+5�
2=3
or
3�−2�=−1
4�+5�=3
•3×3linearsystem
�
1−�
2+2�
3=0
4�
1−5�
2+�
3=−1
�
1+�
2+�
3=1
or
�−�+2�=0
4�−5�+�=−1
�+�+�=1
•4×4linearsystem
�
1−2�
2−7�
3+7�
4=5
�
1+2�
2+3�
3−5�
4=0
3�
1−4�
2−7�
3+�
4=4
2�
1−�
2−9�
3+8�
4=2

6. System of Linear Equations
Definition:
The system (1) is said to be homogeneous linear system if all the constant
terms are zero i.e.
�
1=�
2=.....=�
�=0
Otherwise, the system is said to be non-homogeneous linear system.
Definition:
A solution of the system (1) is a list of numbers for the variables �
1, �
2, . . . ,
�
�,that satisfies all the mequations. The solution is represented in ordered n-
tuples. For example, if a 3×3system has values as: �
1=2, �
2=0, �
3= -3
then solution can be written in ordered 3-tuple as; (2, 0, –3).
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7. Solution of Linear Systems
Note:
For a system of linear equations, we have three possibilities:
•No solution
•Unique solution
•An infinite number of solutions
Question:
Can a system of linear equations has only three solutions or only four
solutions or only 50 solutions?
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8. Behavior of Linear Systems
8

9. Matrix form of Linear Systems
Consider the following system;
(1)
�
11�
1+�
12�
2+...+�
1��
�=�
1
�
21�
1+�
22�
2+...+�
2��
�=�
2
−−−−−−−−−−−−−
−−−−−−−−−−−−−
�
�1�
1+�
�2�
2+...+�
���
�=�
�
In matrix notation, the system (1) can
be written as;
Ax= b
where, A =
�
11�
12....�
1�
�
21�
22....�
2�
⋮ ⋮ ⋮
�
�1�
�2....�
��
x=
�
1
�
2
⋮
�
�
, b=
�
1
�
2
⋮
�
�
9

10. Matrices Associated with Linear Systems
Coefficient Matrix:
The matrix
A =
�
11�
12....�
1�
�
21�
22....�
2�
⋮ ⋮ ⋮
�
�1�
�2....�
��
is called the matrix of coefficients
(or coefficient matrix) of the linear
system (1).
Augmented Matrix:
The matrix
෩??????=
�
11�
12....�
1�
�
21�
22....�
2�
⋮ ⋮ ⋮
�
�1�
�2....�
��
|�
1
|�
2
|⋮
|�
�
is called the augmented matrix of the
linear system (1). It can also be denoted
by ??????
??????. A linear system is completely
determined by its augmented matrix
because it contains all the given
numbers appearing in the system.
10

11. Matrices Associated with Linear Systems
Example:
Consider the following linear system;
5�
1+4�
3+2�
4=3
�
1−�
2+2�
3+�
4=1
4�
1+�
2+2�
3 =1
�
1+�
2+�
3+�
4=0
The above system is non-homogeneous
linear system.
Coefficient Matrix:
A =
504
1−12
412
111
2
1
0
1
Augmented Matrix:
෪??????=
504
1−12
412
111
2
1
0
1
|3
|1
|1
|0
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12. Solutions of Linear Systems
To solve a linear system, we have to follow the four methods given below;
•Cramer’s Rule
•Matrix-Inverse Method
•Gaussian Elimination Method
•Gauss-Jordan Elimination Method
12

13. Cramer’s Rule
Consider the (3×3) system of equations;
�
11�
1+�
12�
2+�
13�
3=�
1
�
21�
1+�
22�
2+�
23�
3=�
2
�
31�
1+�
32�
2+�
33�
3=�
3
We write the above linear system in matrix form as;
Ax= b
where A =
�
11�
12�
13
�
21�
22�
23
�
31�
32�
33
, x=
�
1
�
2
�
3
, b=
�
1
�
2
�
3
13

14. Cramer’s Rule
Cramer’s rule for3×3system is;
�
1=
??????
1
??????
,�
2=
??????
2
??????
,�
3=
??????
3
??????
where ??????=det(??????)≠0and
??????
1=
�
1�
12�
13
�
2�
22�
23
�
3�
32�
33
, ??????
2=
�
11�
1�
13
�
21�
2�
23
�
31�
3�
33
, ??????
3=
�
11�
12�
1
�
21�
22�
2
�
31�
32�
3
Note:
??????
1, ??????
2, ??????
3are obtained by replacing columns 1, 2, 3 respectively (in matrix
A), by the column of matrix b.
14

15. Cramer’s Rule
Example: Solve the following system by Cramer’s rule.
3�
1−2�
2=−1
4�
1+5�
2=3
Solution:
In matrix form, above system can be written as;
A=
3−2
45
, ??????=
�
1
�
2
, �=
−1
3
▪Step-1: ??????≠0
??????=det??????=
3−2
45
=15 –(–8) = 23≠0
15

16. Cramer’s Rule
▪Step-2: Find ??????
1, ??????
2, ??????
3
??????
1=
−1−2
35
= –5 + 6 = 1, ??????
2=
3−1
43
= 9 + 4= 13
▪Step-3: Put the values in formula.
�
1=
??????1
??????
=
1
23
, �
2=
??????2
??????
=
13
23
Therequired solution is (
1
23
,
13
23
).
Note:
If??????=det??????=0, then Cramer’s rule fail.
16

17. Cramer’s Rule
Questions: Solve by Cramer’srule.
12�−5�=23 (2) �
1−3�
2=4
4�+6�=−2 −2�
1+6�
2=5
(3) 3�+4�=14.8
4�+2�−�=−6.3
�−�+5�=13.5
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