# The existence of a linear map onto an affine algebraic set

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!

The bottom arrow is integral and hence induces a surjection from $$mSpec(\Gamma V)\rightarrow mSpec(k[Y_1,..Y_m])$$ Going to the antiequivalent category of varieties we have the commutative diagram
The map $$k^n \rightarrow k^m$$ is a linear map since each co-ordinate in $$k^m$$ namely $$Y_i$$ is a linear combination of the co-ordinates in $$k^n$$ namely $$X_1,...,X_n$$ and $$V \rightarrow k^m$$ is surjective by the discussion above. The argument is now complete.