TITLE: Solving equations using rectangular matrix
QUESTION [0 upvotes]: Suppose I'm solving the following equation and try to find unique values for $v$ that satisfies $Av=u$.
If $A$ is a square matrix and invertible, then $v$ is easily determined as $v=A^{-1}u$.
However if $A$ is not a square matrix, do you know what condition guarantees uniqueness of $v$? For instance, $A$ is $m\times n$, $v$ is $n \times 1$, and $u$ is $m \times 1$, where $m\neq n$. In this case, I'm wondering the condition of a matrix $A$ to guarantee the uniqueness of $v$, if it exists.

REPLY [0 votes]: The equation $Av=u$ has, at most, one solution if and only if the null space of $A$ is $\{0\}$.
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