I encountered this problem in one of my linear algebra homeworks (Linear Algebra with Applications 5th Ed 1.3.44):
Consider a $n \times m$ matrix $A$, such that $n > m$. Show there is a vector $b$ in $\mathbb{R}^{n}$ such that the system $Ax=b$ is inconsistent.
I have a strong intuition as to why this is true, because the transformation matrix maps a vector in $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$, so it is going from a lower dimension to a higher. When the $m$ components in $x$ vary, they would at most be parameterizing an $m$-dimensional subspace in $\mathbb{R}^{n}$. However, my "proof" (which is included below, feels very handwavey and sloppy. It may also be incorrect in a number of places. I'd appreciate it if I could get some pointers on how to formalize proofs of this type a little more, so they are rigorous enough to write on a homework/test, and maybe an example with this example.
My proof:
Consider the case where $A$ has at least $m$ linearly independent row-vectors. Using elementary row operations, rearrange $A$ to $A'$, so these $m$ row vectors are the first $m$ rows. $b'$ will refer to the vector $b$ under the same rearrangement of rows. If we place the first $m$ rows in reduced row ecchelon form using only elementary operations with the first $m$ rows, the augmented matrix $[A'|b']$ will have the following form, where $x_{i}$ is the $i$-th element of the solution vector $x$.
\begin{bmatrix} 1 & 0 & \dots & 0 & x_{1} \\ 0 & 1 & \dots & 0 & x_{2}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & x_{m} \\ a'_{m+1, 1} & a'_{m+1, 2} & \dots & a'_{m+1, m} & b'_{m+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a'_{n,1} & a'_{n, 2} & \dots & a'_{n,m} & b'_{n} \end{bmatrix}
Now consider the $m+1$th row. To eliminate coefficients in this row, it would mean that $x_{m+1} = b'_{m+1} - $$\sum_{i=1}^{m} x_{i}\cdot a_{m+1,i}$, because to eliminate each coefficient would involve scaling by $a_{m+1,i}$ and then subtracting. The system is inconsistent for all $x_{m+1} \neq 0$, so we then choose any $b'_{m+1}$ for which this inequality holds, there are infinitely many, to find $b'$ which makes $A'x=b'$ inconsistent. Then, unswap the rows to make our $b'$ back into $b$ and we have found a vector which makes our system inconsistent.
I would structure your argument like this. First, let $\bar{A}$ be the row-reduced Echelon form of $A$, so that $$\bar{A} = E_1 E_2\cdots E_k A$$ for elementary row operations $E_k$.
1) Can you exhibit a concrete $\bar{b}$ for which the row-reduced system $\bar{A}x = \bar{b}$ has no solution?
2) Can you use the answer to (1) to construct a $b$ for which $Ax=b$ has no solution?