Spaces Associated with a Matrix
There are 3 spaces associated with a matrix.
The Null Space is the set of vectors that solve the homogenous system that the matrix gives the coordinates for.
The Column Space is the set of vectors that are spanned by linear combinations of the columns of the matrix. We will learn 2 methods for finding a basis for the column space. One is to take the matrix to reduced row echelon form and use the ORIGINAL columns where the rref has leading ones. The other is to take the transpose (so columns become rows), take that to row reduced echelon form and use those nonzero rows as they are (which form a basis for the row space of the transpose which is the column space of the original matrix). This basis has more zeros and so would be easier to use. The vectors for the column space will have the same number of elements as there are rows in the original matrix.
The Row Space is the set of vectors that are spanned by linear combinations of the rows of the matrix. We will learn 2 methods for finding a basis for the row space. One is to take the matrix to reduced row echelon form. The rows there form a basis for the row space. They have lots of zeros, so are easier to use than the original rows. The other method is to take the transpose (so rows become columns), take that to row reduced echelon form and decide which ORIGINAL columns (of transpose, so rows of original matrix) form the basis (keep the columns of the transpose where there are leading ones in the rref form). This is a harder basis to work with, but does use vectors specifically from the original set of rows.