Finding the nullity of Matrix A (m x n) I have been given a question to find the nullity of A. 
A is a m x n matrix, what are the possible values of nullity(A)? 
Values given as options are : 
a) (m-1) ≤ nullity(A) 
b) nullity(A) ≥ m 
c) nullity(A) ≤ n 
d) nullity(A)=0 
And all options seems to be true to me. I am sure about options c and d. But options a) and b) are also holding true for few matrix examples. 
For example : 

 A: 
Rank-Nullity theorem:
If there is a matrix $A$  with $m$ rows and $n$ columns over a field, then
  $$Rank ( A) + Nullity (A) = n $$


${}$


Also $Rank(A)\leq min\{m,n\}$

Let $n \lt m$, then $Rank(A) \leq n$
Case I: If in a matrix $A$, $Rank (A)=n$, then $Nullity (A)=0$
Case II: If in a matrix $A$, $0 \leq Rank (A) \lt n$, then $Nullity (A)\leq n$
Let $n \gt m$, then $Rank(A) \leq m$
Case I:  If in a matrix $A$, $Rank (A)=m$, then $Nullity (A)=n-m$
Case II: If in a matrix $A$, $0 \leq Rank (A) \lt m$, then $Nullity (A)\leq n-m$


Although your given example satisfy both the option $a$ and $b$, but from one particular example you can't conclude that those statements are true for all.
A: We know that the nullity of a matrix is the dimension of the space of vectors for which 
$$A\vec x = \vec 0$$
So, for some matrices, the only vector $\vec x$ for which $A\vec x = \vec 0$ is the zero vector itself. In those cases, the matrix has linearly independent column and has $null(A)=0$. This is a very special property, and not all matrices have them.
We know that for an $m\times n$ matrix, the input vector $\vec x$ must have dimension $n$, so the maximum dimension of the space of vectors for which $A\vec x = \vec 0$ is clearly $n$ because $\vec x$ is only $n$-dimensional. 
The nullity of a matrix is hence inclusively bounded between $0$ and $n$.
