Math 601-602 — Spring 2009 (Narcowich)

Homework

Assignment 1

  1. Hefferon's text: Exercises 3.15(c,f), 3.16(b,d), 3.21(b) (pgs. 29-30).
  2. The two systems below, which are given in augmented form, are overdetermined. Either solve them, or show no solution exists.
    3 2 -4 | 1
    2 1 -2 | 3
    1 -1 2 | 1
    1 1 -1 | -2
    
    1  2 | 3
    2 -1 | 1
    1  3 | 4
    -1 2 | 1
    
  3. Put the matrix below in reduced echelon form, using the algorithm given in my Notes on Row Reduction. In addition, find the rank of the matrix.
    -2  1  5  1  2  3
     0  4  9 -1 -6  4
     1 -3  7  0  1 -8
     5  1 -1  1  5  3
    
Due Friday, 30 January.

Assignment 2

  1. The matrix [A|b] below is the augmented matrix for a system of linear equations. Find the reduced echelon form of [A|b], then find rank(A) and rank([A|b]). Is the system consistent? If so, does the system have a unique solution or are there many solutions? State the leading columns of A. Solve the corresponding homogeneous system; put the solution in parametric form.

  2. Repeat the previous question using the complex matrix [C|d] in place of [A|b].

  3. Determine whether the vectors in the set S1 are LI or LD. If they are LD, find a nontrivial linear combination of them that vanishes.

  4. A communications company does an analysis of discrete signals of length five. They find that nearly all signals they encounter can be represented as a linear combination of the three column vectors in the set S2. This allows them to transmit three numbers rather than five, except for the occasional signal that can't be represented this way.

    1. Determine whether the discrete signal s can be represented as a linear combination of the three vectors. If so, find three numbers that represent this signal.

    2. When a signal can be represented in this way, are the three numbers unique? Does it matter? Explain.

    3. Use a stem plot or bar graph to plot the signal and three column vectors.(To create a stem plot, see my notes Discrete signals & Matlab).

  5. Show that the three 4×1 column vectors in S3 are linearly independent. Show that it is impossible to find two vectors u and v such that each of the three vectors in S3 is a linear combination of u and v.

  6. For each of the matrices P and Q, either find the inverse or show that it doesn't exist.

Due Friday, February 6.


Assignment 3

Due Friday, February 13.


Assignment 4

Due Friday, February 20.


Assignment 5

Due Friday, February 27.


Assignment 6

Due Friday, March 13.


Assignment 7

Due Friday, March 27.


Assignment 8

Due Friday, April 3.


Assignment 9

Due Monday, April 13.


Assignment 10

Due Friday, April 24


Assignment 11

Due Friday, May 1