Find the kernel of the linear transformation. (If all real numbers are solutions, enter REALS.)T: P5 → R, T(a0 + a1x + a2x^2 + a3x^3 + a4x^4 + a5x^5) = a0

Accepted Solution

Answer:Kernel is the set of all elements of the form [tex]a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5[/tex]Step-by-step explanation:We are given that a linear transformation T:[tex]P_5\rightarrow R[/tex][tex]T(a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5)=a_0[/tex]We have to find the kernel of the linear transformation if all real numbers are solutions Kernel: It is defined as set of elements whose image is zero.i.e T(x)=0 for any x belongs to domain.To find the kernel of given linear transformation we substituting the given function is equal to zero[tex]T(a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5)=0[/tex] [tex]a_0=0[/tex]Therefore, the basis of  kernel of given linear transformation is K=[tex]\left\{x,x^2,x^3,x^4,x^5\right\}[/tex]Kernel is the set of all elements of the form [tex]a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5[/tex]