I am doing linear algebra and I am a begginer. A thought struck on my mind that I need to work out that what is the relationship between pivots, pivot columns and linear independence?
As far as I perceived, if a matrix is in a row Echelon form, then the first non-zero entry (not necessarily 1) of each row is called its pivot.
Columns that contain the pivots – leading 1's of the rows – are called pivot columns.
Also, I'm aware for the fact that each non pivot column is linearly dependent. Are pivot columns linearly independent?
How pivot works for linear independence?
Best Answer
Pivot columns are linearly independent with respect to the set consisting of the other pivot columns (you can easily see this after writing it in reduced row echelon form).
This means that if each column is a pivot column, all columns are linearly independent. The converse is also true.