University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 7: Due October 24, 2007

1. Define row space, column space and null space.
   
   
2. Define row rank, column rank and nullity.
   
   
3. Show that if R is a row operation (i.e. R=x_ij(c) or R=s_ij or R=h_i(c)) then the row space of RA is equal to the row space of A.
   
   
4. Show that if R is a row operation (i.e. R=x_ij(c) or R=s_ij or R=h_i(c)) then the null space of RA is equal to the null space of A.
   
   
5. Explain how to find a basis of the row space of A.
   
   
6. Explain how to find a basis of the column space of A.
   
   
7. Explain how to find a basis of the null space of A.
   
   
8. Show that (nullity of A) + (column rank of A) = number of columns of A.
   
   
9. Do problems 11, 12, 17, 18, 19, 20 on p252.
   
   
10. Do problems 15, 16, 17, 19 on page 268.
    
    
11. Do problems 9 and 10 on page 282. Also find bases of the row space, column space and null space for each case.