How do you show that rank A = rank $\displaystyle A^{T}$ for any m x n matrix A?
You should know that the coloumn space and the row space of a matrix spam a vector space of the same dimension. The row space of A is equal to the coloumn space of transpose(A) and coloumn space of A is equal to the row space of transpose(A). Therefore, we see that they are the same.
I have to say, (I am studying Linear Algebra at my university now), that a lof of proofs in LinAlg. are a bit different than the proofs in calculus.
At least they feel like it to me.
Like here we simply stated that the columns of A are the rows of A transpose, therefore they have the same dimension.
thanks