# Thread: Find the Kernel of the linear transformations

1. ## Find the Kernel of the linear transformations

I've noticed that most of my problems come, when I try to solve spaces of polynomials rather than real numbers.

T: P3(all polynomials of third degree or less)-->R, T(a+bx+cx^2+dx^3)=a
Find the kernel of the linear transformation.

My approach starts as follows:
According to my book, Elementary Linear Algebra by Larson and Falvo, the kernel of linear transformations is found by equating the right hand side of the T function to 0.
Then, a=0. But, I can't make anything out of it.

Most of the examples are very straight forward and in R. There are no examples on P spaces.

2. ## Re: Find the Kernel of the linear transformations

Yes, that's exactly right. The "kernel" of a linear transformation is the set (a subspace) of vectors that the linear transformation takes to 0. If T(a+ bx+ cx^2+ dx^3)= a, then a+ bx+ cx^2+ dx^3 is in the kernel of T if and only if a= 0. That is the three dimensional subspace of all polynomials of the form bx+ cx^2+ dx^3. The kernel is {{P(x) | P(x)= bx+ cx^2+ dx^3}.

Thanks, Deveno. I have no idea now why I wrote "four"!

3. ## Re: Find the Kernel of the linear transformations

three-dimensional subspace. a four-dimensional subspace would be all of P3 (added just to clarify for the OP, i'm sure it was a typo on your part).