I was having trouble today understanding the correlation between linearly dependent rows and the rank of a matrix. Based on my teaching, we can determine rank by solving for reduced row-echoleon form and simply count all the non-zero rows. This means all the fully 0 rows were linearly dependent on another and cancelled out.
Now, this idea would make much more sense to me if it referred to the columns of the matrix because I always pictured linear transformations as the unit vectors moving to the column vectors' coordinates. If two columns were colinear their span would become a line and thus the matrix output would lose a dimension.
However, reduce row echoleon uses linearly dependent rows to determine rank instead of columns. This idea does not quite make sense to me and I was hoping someone could clarify it based on my understanding.
Thanks