QUESTIONS - CDOG - VECTORS

QUESTIONS - ANSWERS - CDOG - VECTORS

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Can someone please help me with this problem? (u,v) is the inner product general notation. Let u=(u1,u2) and v=(v1,v2) Show that (u,v)=4u1v1+u2v1+u1v2+4u2v2 is an inner product on R^2 by verifying that the four inner product axioms hold. Here are the four axioms: axiom 1: (u,v)=(v,u) axiom 2: (u+v,z)=(u,z)+(v,z) axiom 3: (ku,v)=k(u,v) axiom 4: (v,v)>= 0 and (v,v)=0 if and only if v=0.

THE DEFINITION TO QUALIFY AS AN INNER PRODUCT IS SPECIFIED BY 4 AXIOMS ABOVE AND WE NEED TO SHOW THAT THEY ARE SATISFIED FOR THE GIVEN EXAMPLE.
GIVEN
U = (U1,U2)….AND….V=(V1,V2)…..AND…..
(U,V) = 4U1V1+U2V1+U1V2+4U2V2
TO SHOW THAT
(U,V) IS THE INNER PRODUCT AS DEFINED BY THE 4 AXIOMS.
1.(U,V)=(V,U)
LHS = (U,V)=4U1V1+U2V1+U1V2+4U2V2
= 4V1U1+V2U1+V1U2+4V2U2…BY COMMUTATIVE AND ASSOCIATIVE PROPERTIES OF THE SCALAR COMPONENTS OF VECTORS =(V,U)=RHS

SIMILARLY APPLYING THE DISTRIBUTIVE ,COMMUTATIVE ,ASSOCIATIVE AND SCALAR MULTIPLICATION PROPERTIES OF SCALAR COMPONENTS OF THE VECTORS WE CAN PROVE THE REST AS FOLLOWS.
2.(U+V,Z)=(U,Z)+(V,Z)
LHS = 4(U1+V1)Z1+(U2+V2)Z1+(U1+V1)Z2+4(U2+V2)Z2
=[4U1Z1+U2Z1+U1Z2+4U2Z2]+[4V1Z1+V2Z1+V1Z2+4V2Z2]
=(U,Z)+(V,Z)=RHS
3.(KU,V)=K(U,V)
LHS = 4KU1V1+KU2V1+KU1V2+4KU2V2
=K[4U1V1+U2V1+U1V2+4U2V2]=K(U,V)
4.(V,V)>=0……
(V,V)=4V1^2+V2V1+V1V2+4V2^2
=[V1^2+2V1V2+V2^2]+3[V1^2+V2^2]
=(V1+V2)^2+3(V1^2+V2^2)
SUM OF 2 PERFECT SQUARES WHICH IS ALWAYS>=0

AND…..(V,V)=0
AS GOT ABOVE IF (V,V)=0,WE SHOULD HAVE
V1+V2=0….AND…
V1^2+V2^2=0…WHICH BEING AGAIN A SUM OF 2 PERFECT SQUARES, IS ONLY POSSIBLE IF AND ONLY IF V1=V2=0….THAT IS V=(V1,V2)=(0,0)=0
AS REGARDS THE REQUIREMENT OF IF AND ONLY IF,WE SEE THAT IF
V=0…THEN V1=V2=0…THEN (V,V)=0…ETC…