I do know that the rank of a matrix is the maximum number of linearly independent columns within that matrix whereas the rank of a vector space is the cardinality of its basis. What I do not understand is how the two are related to one another.
Also, why do we arrange the vectors of the spanning set of a given vector space in a matrix and apply row reduction to find the rank/dimension of the vector space. It does make sense to use row reduction when evaluating a system of linear equations (cause you make changes to both sides of the equations and it does not change the outcome ) but why use them to calculate the rank? An example would be of great help.
Thanks in advance 🙂
Best Answer
Treating the rows of the matrix as a set of vectors, the rank of the matrix is equal to the rank (usually called "dimension") of the vector space spanned by those rows. Using the columns instead of the rows will get the same answer.
As for why we use row reduction to find the rank, the key is that the elementary row operations don't change the span of the rows. So once you've row-reduced the matrix, the resulting matrix has the same span as the original. And, since the nonzero rows of a row-reduced matrix are clearly linearly independent, the nonzero rows of the row-reduced matrix are a linearly independent basis for the span of the rows of the original matrix, and the number of such rows is therefore the rank of the matrix.