TITLE: Proof verification
QUESTION [5 upvotes]: I have a problem and a proposed solution. I want to know if I have done it correctly.
Problem Statement: Let $V=F^n$ be the space of column vectors. Prove that every subspace $W$ of $V$ is the space of solutions of some system of homogeneous linear equations $AX=0$.
My solution: The null space of an $m$ x $n$ matrix $A$, written as $Nul A$, is the set of all solutions to the homogeneous equation $Ax = 0$. The null space of an $m$ x $n$ matrix $A$ is a subspace of $R^n$. Equivalently, the set of all solutions to a system $Ax = 0$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $R^n$. 
Then I proceeded to prove that the properties of a subspace hold for the null space, and hence the null space is a subspace of $R^n$. Please tell me if this is right... this problem has been on my head all day!!
Thanks!

REPLY [3 votes]: Let $W$ be your subspace. If $W$ is all of the space, it is the kernel of the null matrix.  If it is $\{0\}$, it is the kernel of any nonsingular matrix. So assume your subspace is proper, and let $\{w_1,\ldots,w_s\}$ be a basis. Extend this to a basis $\{w_1,\ldots,w_s,w_{s+1},\ldots,w_n\}$ of your space, let $\{w_{s+1},\ldots,w_n\}=W'$. Define the transformation $f:V\to V$ as the projection onto $W'$ through $W$. Then $W=\ker f$, and you can consider the matrix of $f$ in the canonical basis to get an appropriate system of equations. 
Alternatively, use the canonical inner product: if $W$ is your subspace, consider $W^\perp$. Then use that $W=(W^\perp)^{\perp}$ to get a set of equations for $W$, namely of the form $w_i\cdot x=0$ for $i=1,2,\ldots, r$ where $\{w_1,\ldots,w_r\}$ is a base for $W^{\perp}$.