Prove: Every Subspace is the Kernel of a Linear Map between Vector Spaces

Question

I'm trying to show that, given two finite-dimensional vector spaces $V,W$, and any subspace $V'$ of $V$, that there is a linear map $T:V\to W$, whose kernel is precisely $V'$, given the condition that $\dim V-\dim(\ker T)<\dim W$. I would like to know if the same is true for infinite-dimensional spaces.

Because of Rank-Nullity, we have restrictions on the respective dimensions; we need

$$\dim W =\dim V-\dim(\ker T), \qquad\mbox{ (I think) }.$$

This is my work: let $\dim V=m $, $\dim W=r$; $r=m-n $, for $\dim(\ker T)=n$. We start by taking a basis

$$B_V':=\{v'_1,\ldots,v'_n\},$$ and extend $B_V'$ into a basis $B_V:=\{v'_1,\ldots,v'_n,v'_{n+1},\ldots,v'_m\}$ for $V$. Let $B_W:=\{w_1,w_2,\ldots,w_r\}$.

Now, we define $T$: $$T(B_V'):=0,$$ i.e., $T$ is zero for every vector in $B_V'$, and $T$ is linear. By linearity, $T$ is zero on $V'$.

Now:

This is the part that seems harder: how to define $T$ outside of $V'$, so that $T(w) \neq0$ for $w \in V\setminus V'$.

My idea is:

i)We set up a bijection between the basis vectors in $B_V\setminus B_V'$, and the basis vectors in $B_W$, say:

$$T(v'_{n+1})=w_1,$$ $$T(v'_{n+2})=w_2,$$ $$\vdots$$ $$T(v'_m)=w_r,$$ and extend $T$ linearly.

ii) Since a bijection between basis vectors extended linearly gives rise to a Vector Space isomorphism, the kernel of $T|_{V\setminus V'}\rightarrow W$ is an isomorphism, so that its kernel is $0$.

Does this work? Can we extend it to the infinite-dimensional case?

Thanks.

Answer 1

As the OP asked for a method using quotient spaces, here we go.

Let $V$ be a vector space (no restriction on dimensions!), and consider a subspace $U\subset V$. We can form the quotient space $V/U$: by definition, as a set, it is the set of equivalence classes $v+U$ where $v+U=u+U$ if and only if $v-u\in U$. There is a natural notion of addition (we add the representatives) and a natural notion of scalar multiplication (we multiply the representative by the given scalar) making the set $V/U$ into a vector space.

Now, define the projection map $$T:V\to V/U$$ by $T(v)=v+U$. This is well defined (exercise!), and the kernel is precisely $U$.

One advantage of this method is that it is more tidy: no bases. Also, it does not depend on the dimension of your space.

Answer 2

Let the spaces be finite dimensional. If $\dim(W) \ge \dim(V)-\dim(V')$, then it can be done.

Extend, as you did, a basis $v_1',v_2',\dots, v_m'$ for $V'$ to a basis $v_1',\dots,v_m', v_1,\dots, v_n$ for $V$. Let $\{w_1,w_2,\dots, w_n\}$ be a linearly subset of $W$, and map the $v_i'$ to $0$, and $v_j$ to $w_j$, and extend by linearity. Because of the linear independence of the $w_j$, the kernel of the resulting linear transformation is $V'$.

For infinite dimensional spaces, then, assuming the Axiom of Choice, the same argument works. Since subtraction is problematical, the appropriate condition is that $\dim(V/V') \le \dim(W)$.