Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix?

Question

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the proof, but I don't "get it". Can someone help me out with this ? I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level.

Answer 1

Strictly an intuitive answer, but you know that if you are solving a problem in Rⁿ, you will need n variables and n independent equations. You know this from Algebra 2.

The variables are represented by the column vectors and the equations by the row vectors. If you have three independent equations, you need 3 variables that make them work and your answer lies in R³; if you have 4 variables, and three equations either one variable is irrelevant because you are in R^3, or you need a fourth equation because you are in R⁴.

Any additional data in the form of irrelevant data (x,y,z,w in a problem that uses x,y, and z, but not w), or superfluous (dependent) equations simply don't change the rank. A rank 2 matrix answers problems in R², rank 3 in R³.