Math 210-01: Linear Algebra: Reading Homework 4.6

1. Matrix Rank and Systems : with what is this section concerned?
   
   1. Row and Column Spaces : what are the row and column space of a matrix?
      
      1. row spaces : what do we know about the row spaces of row-equivalent matrices? how are the rows of these related to the spaces?
         
      2. finding row spaces and bases : how can we find the row space of a matrix? how can we find a basis for the subspace of Rⁿ spanned by a set of vectors?
         
      3. column spaces : how can we find a basis for the column space of a matrix?
         
      4. dimension : how are the dimension of the row and column spaces of a matrix related?
         
   2. Rank : what is the rank of a matrix? how can we find the rank of a matrix?
      
   3. Nullspace : what is the nullspace of a matrix? how is it related to the nullity of the matrix? how is it a solution space?
      
      1. rank, nullity, size : how are the dimensions of a matrix related to its nullity and rank?
         
   4. Nonhomogeneous Systems : how are solutions to a nonhomogeneous system related to the nullspace of a the coefficient matrix?
      
      1. column space : how is the column space of a coefficient matrix related to solutions of a nonhomogeneous system?
         
   5. Square Systems : what conditions are equivalent when we discuss ``square'' linear equations?