Define T P3 R3 by Tp p1 p2 p3transpose T038Transpose Calc

Define T : P3 R^(3) by T(p) = (p(1), p(2), p(3))^(transpose) . T=(0,3,8)^(Transpose)

Calculate dim(ker(T))

Find a basis for ker(T)

Let, p be in kernel of T

let,p=a+bx+cx^2

So,

p(1)=0=a+b+c+d

p(2)=0=a+2b+4c+8d

p(3)=0=a+3b+9c+27d

SUbtracting first two equations gives

b=-3c-7d ------------------ A

Subtracting second and third equations gives

b=-5c-19d ----------------------B

Substracting first and third equation gives

b=-4c-13d --------------------------C

A-B gives

c=-6d

Substituting in A gives

b=11d

Substituting in first equation gives

a+b+c+d=0

a+11d-6d+d=0

a=-6d

Substituting in second equation gives

0=a+2b+4c+8d

0=-6d+22d-24d+d=0

-6d-d=0

Hence, d=0

Hence, a=b=c=d=0

Hence, only element in ker(T) is the 0 polynomial

dim(ker(T))=0

basis for ker(T)={0}