This document highlights important concepts and fundamental tools needed to understand Linear Algebra. It covers essential terms and vocabulary was well as syntax needed to apply the concepts. Moreover, it is organized by topic so that it is easy to navigate and use.
,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
⑪
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