Let $K$ be an algebraically closed field, $X\subset K^n$ be an affine algebraic set and $I$ be the ideal generated by all polynomials in $K[x_1,\cdots,x_n]$ that vanish on $X$.
Prove that there exists a linear subspace $L$ of $K^n$ and a linear map of $K^n$ onto $L$ that maps $X$ onto $L$.
I have already proved the hint, which states that if $B/A$ is integral and $K$ is an algebraically closed field then a ring homomorphism from $A$ to $K$ can be extended to $B$ to $K$. My thought of this problem is to use Noether's normalization theorem, so there exists $K[y_1,\cdots,y_m]\subset K[x_1,\cdots,x_n]/I$ where the extension is integral and $y_1,\cdots,y_m$ are linear combinations of $x_1,\cdots,x_n$. Then $L$ should be this $m$-dimensional subspace of $K^n$. But I am not sure how to construct the map. Thanks for any help!