MA 35100 Lecture Notes - Lecture 17: Row And Column Vectors
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Given an (m, n) matrix a, the rank (a): dimension of rowspace (a): #ofpivots in an eschelon from. Dimension of colspace (a) = rank (a) = rank (a) t. "flipping rows and columns rowspace (a) = colspace (at) aim rca) = aimcol (at) = dimr (at) =rank at = rank a. Theorem 2. 15: all the columns of a are linearly independent, dim((a) = 4 = rank (a), so min. Since aim (a) = dimrca), there must be at least as as many rows as columns. All the rows of a are linearly independent, dimr(a) = m =rank a, son i m. Examine ax = b where a cm 1m, n) (r). This system is called solvable ifthere is a solution x that verifies the equation. If, for anybin km, the system is @solvable then rank (a) = m. Reversible: if rank (a) = m, the system is solvable for any b.
