This throws me off track completely - its like pushing me out of moving train.I am referring to page 65 of Shilov (Linear algebra).
The author clearly states that in Homogeneous system of linear equations:
- If the coefficient matrix has order k x n (k -> number of equations, n -> number of unknowns)
- r is rank of the matrix.
Then linear solution space has dimension n-r.
I would think if rank is r, then the number of linearly independent rows is r. So if xi denotes any solution for r+1st equation to kth equation, it should be totally describable by linearly independent solutions ri from i=1 to i=r (Again going by rank).
Yet, the dimension of solution space L is given by n-r.
Update: What's being said is starting to makes sense. I am probably mixing up the concept of linear independence of columns of basis minor matrix (rank r) with dimension of solution space.
I guess the fact that first r columns of coefficient matrix of rank r are linearly independent implies that other columns r+1 .... n is expressible in terms of r columns.
That gives us freedom to arbitrarily choose solutions cr+1...n for dependent column variables with solutions c1...r for linearly independent column elements uniquely determinable (By Cramers rule). This "freedom" manifests as dimension n-r of solution space. Does the above sound coherent?