Applied Linear Algebra

Quiz 2

Due 25 March 1999

This is a take-home group quiz. A group should consist of 3-4 students. One solution set per group is accepted.

Cite all major theorems and show all relevant work. Please do each question on a separate page.

  1. Let A and B be square matrices of the same size. Prove, or find a counterexample to the following statement:


    2. displaymath16

  2. Let V be the vector space over the field of real numbers tex2html_wrap_inline20 spanned by functions tex2html_wrap_inline22 and tex2html_wrap_inline24 where tex2html_wrap_inline26 is an independent variable. Let A be a mapping tex2html_wrap_inline30 defined by the following:


    3. displaymath32

    Prove that A is a linear mapping, find its range and null-space. Does the inverse mapping exist? Find a general solution to the equation:

    displaymath36

  3. Let V be the vector space over the field of real numbers tex2html_wrap_inline20 of all polynomials with complex coefficients of one complex variable x having degree at most 2. Let A be a mapping tex2html_wrap_inline46 defined by the following:


    4. displaymath48

    Prove that A is a linear mapping, find its range and null-space. Does the inverse mapping exist? Find a general solution to the equation:

    displaymath52

  4. In the previous problem, introduce a basis of V and write down the matrix, corresponding to A for this basis. Find the rank of the matrix.

    5.  

     


Andrew Knyazev

Fri Mar 5 09:34:06 MST 1999