Introductory Linear Algebra Notes for Beginners

Intro to linear algebra notes




Samikhas

,Chapter 1 : Matrices , Vectors, & Systems of Lincar Equations :


.1
1
Matrices & Vectors



Matrix :
rectangular array
of real numbers
-

↓
Scalars (R)
m i ro u s - i
Size
-
of matrix (MXn) Nicolumns + j
-




Square (m = n) [aij bij] =




:
B :
[i]
9
and
[ ]
b2 = 8 and 112 :


-




matrices]
BFC
So , Bas a re 3x2 inventory matrices




Submatrix :
deleting from M entire rows ,
colums, or both




(i
:
Ethmetic use of Matrix :


Sum of A & B (A + B) :
aij + bij



n + B =

[i]
an

[s]
=




Subtraction (A-B) :
Rij-bij




A B

[=&
-
=




~ is of Matrix Addition & Scalar Multiplication : (PROOFS) :




(a) A + B B + A commutative l aw of matrix addition




↳
=




(b) (A + B) + c = A + (B + ) ) associative l aw of matrix addition
-
in re s


Ex : 4 A + B + c
(c) A + 0 = A




(d) A + 1 -
A) = 0




(e) (s + (A = S( + A)


(f)s(A + B) = SA + SB


(g)(s + z)A = SA + tA

, Transpose of my n matrix A :




n X m matrix



denoted AT (i , j) + Li, i)




·
[& and c =

[488]
Properties of Transpose :




(a) (A + B)T = AT + BT



(b) (SA)T = SAT


(c) (AT)T =
A




Vectors :

Row ve c tor : matrix with I row




Column vec tor : matrix with I column



~
Components : entries of vec tor




Vector Representation U
: & v




[G ] :-




can be added & Scalars
multiplied by

Y vec tor addition & Scalar multiplication




[a]
~ A is an my n matrix



vi :
vi commen




Geometry of Vectors :




v= [5 ] rec tor in R&




Ga ,
b




Ex : A boat cruises in still wa te r toward the nor theast at 20 miles per hour . The velocity of the boat is a ve c tor that points i n the direction



Of the boat's & whose is 20 the boat's speed . If the positive y-axis north and the positive X-axis
motion, length , represents represents



East, the boat's direction makes an
angle of 450 with the X-axis .




~ [in] S
41




42
=




=
20105450


209in 450
:




=
102


102
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