Math 060 100917 The Rank Theorem

Definitions: column space, row space, rank. Observe: row equivalent matrices have identical row spaces. Finding the basis of a row space via Gaussian elimination (or Gauss-Jordan reduction). Rank = number of pivots in row-echelon form. Nullity = # of free variables. Rank theorem. Exercise using the rank theorem: if rank is full, then Ac = Ad implies c = d. Theorem: dimension of column space = rank(A). Observation: columns corresponding to pivots form a basis of the column space. Example.