# Show that for any matrix $A_{m \times n}$ , the row rank and column rank are equal [duplicate]

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 .

## 1 Answer

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.

• Im fairly new to the whole concept of a rank , If its not too much trouble could you please tell me what 'number of linearly indepedent columns' means ? – Nikhil Kapoor Sep 30 '16 at 12:17
• Every column in matrix represents a vector. Two vectors are linearly independent if one is not a multiple of another, in other words, u,v are linearly independent if there is no nonzero scalar a such that u=a*v. Set of vectors is linearly independent if none of vectors from that set can be expressed as a linear combination of others. Now, rank of column space is the number of linearly independent column vectors. For example, rank of (0,1,0), (0,2,0),(1,0,0) is 2 because second vector is 2*(0,1,0) so it is linearly dependent with other vectors. – MaliMish Sep 30 '16 at 13:25