Theorems from Rank-nullity

For a matrix $A$ over a scalar field $\mathbb F$ with $n$ columns,

Let $A$ be a matrix over an infinite scalar field $\mathbb F$ (such as $\mathbb R$ or $\mathbb C$), let $b$ be a vector of the appropriate dimension and let $[A\mid b]$ be the augmented matrix formed by appending $b$ to $A$. The equation $Ax=b$ has:

1. no solution if $$\operatorname{rank}(A)\ne\operatorname{rank}([A\mid b]);$$
2. a unique solution if $$\operatorname{rank}(A)=\operatorname{rank}([A\mid b]) \quad\text{and}\quad \operatorname{nullity}(A)=0;$$
3. infinitely many solutions if $$\operatorname{rank}(A)=\operatorname{rank}([A\mid b]) \quad\text{and}\quad \operatorname{nullity}(A)>0.$$

In the last case, if $\operatorname{nullity}(A)=k$, then every solution has the form

where $x_p$ is any particular solution of $Ax=b$, $\{u_1,\ldots,u_k\}$ is a basis for $\ker(A)$ and $\omega_1,\ldots,\omega_k\in\mathbb F$.

If $T:V\to W$ is a linear map over a scalar field $\mathbb F$ and $V$ is finite-dimensional, then