1. ## basis for kernel

I know that a basis for the kernel is the map of the nullspace of the standard matrix for the linear transformation.

I found the nullspace for a particular problem to have a basis formed by the set
1/2
-3
-1/4
1

The original basis was the standard basis {1, x, x^2, x^3}.

Because I am working with polynomials, my professor said that the basis for a kernel must be polynomials. How do I change the nullspace basis given above into a polynomial?

2. Originally Posted by PvtBillPilgrim

I found the nullspace for a particular problem to have a basis formed by the set
1/2
-3
-1/4
1

The original basis was the standard basis {1, x, x^2, x^3}.

Because I am working with polynomials, my professor said that the basis for a kernel must be polynomials. How do I change the nullspace basis given above into a polynomial?
The matrix that you got is the coordinate vector. Meaning the coordinate vector relative to the ordered base:
(1,x,x^2,x^3)
That means the basis is,
(1/2)(1)+-3(x)+(-1/4)(x^2)+1(x^3)
Thus, the vector (polynomial in this case) is,
$\frac{1}{2}-3x-\frac{1}{4}x^2+x^3$
This is the basis for the nullspace.