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]
