Orthogonal complement of $\operatorname{im}(A)$

Question

If I have a $n\times d$ matrix $A$, $$\dim(\operatorname{im}A)^\perp +\dim(\operatorname{im} A)=d$$ since $\operatorname{im}(A)$ is a subspace of column space of $A$. Also, $$\dim(\operatorname{im}A^t)^\perp + \dim(\operatorname{im}A^t) = n$$ since $\operatorname{im}(A^t)$ is a subspace of row space of $A$. Are these two statements correct?

Answer 1

Yes, both of those statements are correct. In general, if you have $W$ a subspace of finite-dimensional inner product space $V$, then $$\dim W + \dim W^\perp = \dim V.$$ This implies both of the statements given.