Vector in R space.

Tuple = A tuple is a finite ordered list of elements sequence of n-elements where n is a non-negative integer.

Set of n-tuples of Real Numbers => Rn

Note: O-tuple = Empty space 
Eg. (0,0,0,0...........)
A particular n tuple in Rn 

U = ( a1,a2,a3,..................an ) is called a point or vector.
[ where, (a1,a2,a3,.............an) is called Scalar & U is called Vector.]

                                          or,

U = [a¡] ,where i is called as components/entries/co-ordinates/elements.

Note: Elements of R = Scalar .

1.) Two vectors U & V are equal i.e U=V if they have same number of components and the corresponding components are equal.
 e.g.
 U =[a1,a2,a3]
 V =[b1,b2,b3]
 If U =V then
  a1=b1 , a2=b2 , a3=b3

2.) U=[0,0,0..........0]
This is Zero vector and is denoted as Bold 0 (𝐨).

Example :

Q) Identify the space : 

a) (2,-5) -> R2 


b) (3,4,5) -> R^3


Q) Find x,y,z such that (x-y , x+y , Z-1)=(4,2,3).

soln:

x-y=4          (i)
x+y=2         (ii)
z-1=3         (iii)
From equation (iii) we will get, Z=4
Solving equation (i) & (ii),we will get,
             2x=6 => x=3
putting value of x=3 in equation (i),we will get
              y=-1
So,  X=3,y=-1,Z=4
In vector form (3,-1,4).

COLUMN VECTORS :

Vector is n-space Rn is written horizontally as well as vertically.
eg. 
Types of matrices - with Definition and Examples - Teachoo
This is R^4. (Vector in 4 space )

Note: Operation defined for row vectors is analogous for column vector.
eg.
1.) U=[1,2] , V=[3,4]
      U+V=[4,6]

2.)
Beginner's Introduction to Matrices - Towards Data Science
Row and  Column matrix addition is not possible.

VECTOR ADDITION:

Suppose two vector U = (a1,a2,a3............an) & V = (b1,b2,b3............bn) U & V are in same Rn .

So,U+V=( a1+b1,a2+b2,a3+b3...................an+bn)

SCALAR MULTIPLICATION:

Let k be any Real number.
       U=(a1,a2,a3,..............an) in Rn   .  

     kU=(ka1,ka2,ka3.............kan)

Note: Negatives & Subtraction are defined in Rn as follows :
               -U =(-1)U. => Negatives of U.
                U-V=U+(-V) => Difference of U & V.

LINEAR COMBINATION OF VECTORS:

Let U1,U2,U3...............Un are vectors in Rn & V1,V2,V3.........Vn are vectors in Rn .

M=U1V1 + U2V2 + U3V3 +..............+UnVn.
(Where M is Linear Combination of vectors.)

Note: Zero vector 0 =(0,0,0..............0) in R is similar to scalar 0 in R.

Example :

Q) Take U=(2,-3,6) ,V=(9,2,-8) Find :

(i) U+V  (ii) 7U  (iii) -V   (iv) 5U-3V

Soln:

(i) U+V = (2+9,(-3)+2,6+(-8)) => (11,-1,-2)

(ii) 7U = 7*(2,-3,6) => (14,-21,42)

(iii)-V = -1*(9,2,-8) => (-9,-2,8)

(iv)5U-3V = 5*(2,-3,6) - 3*(9,2,-8)
                  = (10,-15,30) - (27,6,-24)
                  =((10-27),(-15-6),(3-(-24)))
                  =(-17,-9,27) Ans...

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