My book has asked the following question:
Let $A$ be an $m×n$ matrix.
Suppose dim Nul(A) = 3
Give a formula for the rank of A in terms of m or n.
Now, I know that the formula is (what I have been taught is):
rank (A)+ dim Nul(A)= n
But, the question says in terms of m or n, and I know only how to express in terms of n, which I have shown above. This got me thinking, is there an acceptable way of writing up a formula in terms of m?
No. Adding all-zero rows to a matrix $A$ increases the value of $m$, but the rank stays the same, and the null space stays the same. Good question, though.