Linear Algebra: Check my work? Task: Find a matrix where the kernel is a subspace of the range. My initial thought was highly simplistic, which makes me double back and think twice. I would love it if someone could point out any mistakes or tell me if I'm right. The matrix I had in mind:
$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
In this case, we see that the kernel is made up of a single vector, namely the zero vector, as that's the only solution that maps to the zero object. Next, we see that the columns are made up of linearly independent vectors. This means that the column space spans the entire $\mathbb R^3$. The zero vector can be easily shown to be a subspace of the vector space.
 A: maybe easier to think in terms of a basis, say $e_j$ with $1 \le j \le n$. then consider a map $T$ satisfying:
$$
Te_1 =0 
$$
and for $k \gt 1$
$$
Te_k = e_{k-1}
$$
A: What about this:
Let
$$
\mathbf{B}
=
\begin{bmatrix}
     1&     0\\
     0&     1\\
     0&     0\\
\end{bmatrix}
\quad \text{and} \quad
\mathbf{C}
=
\begin{bmatrix}
     0&     0\\
     1&     0\\
     0&     1\\
\end{bmatrix}.
$$
Does $\mathbf{A} = \mathbf{B}\mathbf{C}^{T}$ have the desired property?
More importantly, is the rationale behind this example clear? 
We want to find a matrix $\mathbf{A}$ such that its nullspace is a subspace of its range. 
But the nullspace is a set of vectors orthogonal to the row space of $\mathbf{A}$ (or equivalently orthogonal to a basis of its row space).
Lets say that I choose first a basis for the row space. Let $\mathbf{C}^{T}$ be that choice as in the example above. Then $[1\, 0\, 0]^{T}$ will be in the nullspace of $\mathbf{A}$. And in fact it spans the entire subspace (because in my example $\mathbf{C}$ has rank $2$).
Now, we need to make sure that $[1\, 0\, 0]^{T}$ is in the range of $\mathbf{A}$. It suffices to ensure that a basis of the nullspace is used to form a basis of the column space of $\mathbf{A}$.
Here I selected this basis to be the columns of $\mathbf{B}$ (which includes $[1\, 0\, 0]^{T}$). 
Finally note that it is also important that both $\mathbf{B}$ and $\mathbf{C}$ are full rank.
