Can somebody first please tell me what is the row rank and column rank of a matrix ? What is the relation of each with the rank of a matrix ? Any kind of explanatory proof would be very helpful , thanks ! Also , I am not looking for an intuitive proof , I need the mathematical proof and its explanantion , for I did not uderstand much when I looked at the proof on the internet .
Row rank: number of linearly independent rows of matrix.
Column rank: number of linearly independent columns of matrix.
Rank of matrix is the number of linearly independent rows (or columns, since these two are the same).
Every matrix $A$ has same number of pivot elements as its transpose matrix $A^T$. Since row rank$(A)$ = row rank$(A^T)$ = column rank $(A)$, row rank of matrix is same as column rank of matrix.
For more explanations on pivot elements see (Reduced) Row Echelon Form of matrix.