Algebra Master Notes (75 Pages) – All Formulas, Theorems & Solved Problems for Easy A+ in Exams

Algebra Lecture Notes
The 14 lectures will cover the material as broken down below :

1-3: Linear Systems, Matrix Algebra
3-4: Inverses and Transposes
4-5: Vector Spaces and Subspaces
6: Bases
7: Dimension
8: Dimension and Subspaces
9-10: Linear Maps. Rank-Nullity Theorem
11-12: Matrices representing Linear Maps
13-14: Inner Product Spaces

,Liner Systems and Matrices



1 2


11 1 + 12 2 + + 1 = 1;
21 1 + 22 2 + + 2 = 2;



1 1 + 2 2 + + =


( 1 2 )



( b)

11 12 1 1
21 22 2 2
= b=

1 2


b




3 + 2 = 2; + + = 2; 2 +4 + =0

=2


3 + 2(2 )=5 +3 4= 2 = 5 + 3 = 2;
2 + 4 + (2 )= +3 +2=0 = +3 = 2

4 =4

=1 =( 2 ) 3= 1 =2 =2

, ( ) = (1 1 2)


(1 1 2)




3 1 2 2
1 1 1 2
2 4 1 0




1 1 3 1
2 + =3 2
+ 2
= 2 2




( ) =0

= ( )

, 12

3 1 2 2 1 1 1 2
12
1 1 1 2 3 1 2 2
2 4 1 0 2 4 1 0
12 ( 3)
13 ( 2)

1 1 1 2 1 1 1 2 1 1 1 2
12 (¡3) 13 (¡2)
3 1 2 2 0 2 5 8 0 2 5 8
2 4 1 0 2 4 1 0 0 2 1 4
2 2( 1 2)
1 1 1 2 1 1 1 2
2 (¡1 2)
0 2 5 8 0 1 2 12 4
0 2 1 4 0 2 1 4
21 ( 1)
23 ( 2)
1 1 1 2 1 0 1 12 2 1 0 1 12 2
21 (¡1) 23 (¡2)
0 1 2 12 4 0 1 2 12 4 0 1 2 12 4
0 2 1 4 0 2 1 4 0 0 6 12
6 3( 1 6)
1 0 1 12 2 1 0 1 12 2
3 (¡1 6)
0 1 2 12 4 0 1 2 12 4
0 0 6 12 0 0 1 2
2 12 32 ( 2 12 )
1 12 1
31 (1 2 )

1 0 1 12 2 1 0 1 12 2 1 0 0 1
32 (¡5 2) 31 (3 2)
0 1 2 12 4 0 1 0 1 0 1 0 1
0 0 1 2 0 0 1 2 0 0 1 2
=1 = 1 =2



12 12 ( 3) 13 ( 2)
(1 0 0)
1 (1 3) 3 12 ( 1) 13 ( 2)
1
2( 1 2) 21 ( 1) 23 ( 2) (0 1 0)
(0 0 1)