Linear Algebra I: Homework 7

Due: Friday, October 13, 2017
  1. Consider the set of vectors in :

    1. Find a vector in which can be expressed as a linear combination of the other vectors in .

      Any vector is a fine answer here. For example,

    2. Make a new set of vectors by removing your vector from part (a) from . Is linearly independent? Explain.

      Based on our answer to (a), we’d get

      This set isn’t linearly dependent, since,

    3. Find a vector in which can be expressed as a linear combination of the other vectors in .

      From our discussion in (b)

    4. Make a new set of vectors by removing your vector from part (c) from . Is linearly independent? Explain.

      Based on our answer to (c), we’d get

      which is a linearly independent set of vectors since the matrix,

      has reduced row echelon form

      that has a pivot in every column.

  2. A unit vector is a vector whose magnitude is 1.

    1. Describe all unit vectors in .

      There are a few different ways to express this answer. One way is any vector of the form,

      for any .

    2. For which unit vectors is

      a basis for ?

      Any answer to part (a) except

      works, since the determinant of the matrix

      is . That is, it is a basis so long as .

    1. Find a basis for the vector space of diagonal matrices.

      There are many correct answers. One standard one is,

    2. An upper triangular matrix is a matrix whose entries below diagonal entries are all 0. Find a basis for the vector space of upper triangular matrices.

      There are many correct answers. One standard one is,

  3. Consider the two bases for ;

    1. Find a matrix that changes column vectors for basis into column vectors for basis .

      The matrix,

      changes -vectors into standard vectors, and the matrix

      changes -vectors into standard vectors. On the other hand, changes standard vectors into -vectors.

      So the matrix changes -vectors into -vectors.

    2. Find a matrix that changes column vectors for basis into column vectors for basis .

      We want to do the opposite of what we did in (a). This means, we want the inverse of our answer: is the matrix that changes -vectors into -vectors.

  4. Is 5 an eigenvalue of the matrix:

    Explain your answer.

    Yes, it is. This is because, if we plug in into the characteristic polynomial of the matrix, we get the determinant of the matrix,

    which is zero.