Kelvin function? I meant the kernel lol.

1) If

then

meaning

. But the

are linearly independent so the

are identically zero.

2) Let

be a basis for the vector space. Let

be a vector in the kernel of T. Then

. But because

are linearly independent, this implies that the

are identically zero so x=0. So the kernel of T is zero. It follows that T is injective. Now show it is surjective and use these to deduce it is invertible.

P.S. I use the Einstein summation convention. I'm going to get in trouble if I don't get back to work now lol. Good luck!