University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 8: Due November 2, 2007

1. Define linear transformation, kernel and image.
   
   
2. Let L be a linear transformation. Explain carefully and precisely what the matrix of L with respect to bases S and T is.
   
   
3. Suppose that L is a linear transformation from R^m to Rⁿ. Explain carefully and precisely what the standard matrix of L is.
   
   
4. What are similar matrices?
   
   
5. Let A be the matrix of L with respect to bases S and T. Explain carefully and precisely the relationship between the kernel of L and the null space of A.
   
   
6. Let L be a linear transformation. Let A be the matrix of L with respect to bases S and T. Explain carefully and precisely the relationship between the image of L and the column space of A.
   
   
7. Let L be a linear transformation. Explain how to find a basis of the kernel of L.
   
   
8. Let L be a linear transformation. Explain how to find a basis of the image of L.
   
   
9. Do problems 1, 2, 7, 8, 13, 14 on p. 372-373.
   
   
10. Do problems 1, 2, 3, 16 on p. 387-388.
    
    
11. Do problems 3, 5, 7 on p. 398.
    
    
12. Do problems 2, 5 on p 413.