Scribd
Upload
26 views5 pages

Inner Product Space: Def Dot Product:-For X, y R, The Dot Product of X&y

Uploaded by

Muhammad Farid Shahid
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPTX, PDF, TXT or read online on Scribd
26 views5 pages

Inner Product Space: Def Dot Product:-For X, y R, The Dot Product of X&y

Uploaded by

Muhammad Farid Shahid
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPTX, PDF, TXT or read online on Scribd
You are on page 1/ 5

INNER PRODUCT SPACE

Def Dot Product :-


For x,y € Rⁿ , the dot product of x&y,
denoted x.y is defined as

x.y=x1y1+x2y2+…..xnyn

Where x=(x1,x2,…xn) and y=(y1,y2,…yn)


Note that Dot Product of two vectors in Rⁿ is a number not a vector. Obviously,
x.x=||x||² for all x €Rⁿ. The dot product on Rⁿ has following properties:

◦ x.x ≥ 0 for all x € Rⁿ.


◦ x.x = 0 if and only if x = 0.
◦ For y € Rⁿ fixed , the map from Rⁿ to R that sends x € Rⁿ to x.y is linear.
◦ x.y = y.x for all x,y € Rⁿ.
Def INNER PRODUCT :-
An inner product on V is a fiction that takes each ordered pair (u,v) of
elements V to a number <u,v> € F and has following properties :-

◦ Positivity:
< v,u > ≥ 0 for all v € V

◦ Definiteness :
<v,v> = 0 if and only if v=0

◦ Additivity in first Slot :


< u+v,w> = <u,w> + <v,w> for all u,v,w € V

◦ Homogenity in first slot :


<^u,v> = ^ <u,v> for all ^ € F and all u,v €V

◦ Conjugate Symmetry :
<u,v>=<v,u> for all u,v € V
Properties of INNER PRODUCT :

◦ For each fixed u €V , the function that takes v to <v,u> is a linear map from V to F.
◦ <0,u> = 0 for every u € V.
◦ <u,0> = 0 for every u € V.
◦ <u,v+w> = <u,v> + <u,w> for all u,v,w € V.
◦ <u,^v> = ^<u,v> for all ^ €F and u,v € V.
Def Norm :
For v€V , the norm of v , denoted ||v|| is defined by

||v||=√<v,v>

Basic Properties :
Suppose v€V

◦ ||v|| = 0 if and only if v=0.


◦ ||^v|| = |^| ||v|| for all ^€F.

You might also like