## linear change of coordinates preserving a certain property

Doesn't a linear change of coordinates preserve complete intersection for a set of homogeneous polynomials, all of the same degree, in a polynomial ring?

That is, apply a change of coordinates to a set of homogeneous polynomials {f_1,... f_k} in C[x_1,...,x_M] to obtain {h_1,..., h_k}. Suppose now that the variety cut out by {h_1,...,h_k} is a complete intersection. Doesn't this imply that the original set of generators {f_1,... f_k} formed a complete intersection?

This seems very plausible.
 Never mind. Please disregard this post.

 Tags algebraic geometry