Page 239 - 35Linear Algebra

12.4 Review Problems 239

3
3
4. Let L be the linear transformation L: R → R given by
   
x x + y
y
L   =  x + z  .
z y + z

Let e i be the vector with a one in the ith position and zeros in all other
positions.
(a) Find Le i for each i = 1, 2, 3.

 1 1 1 
m 1 m 2 m 3
(b) Given a matrix M =  m 2 1 m 2 2 m 3 2  , what can you say about
m 3 1 m 3 2 m 3 3
Me i for each i?
(c) Find a 3 × 3 matrix M representing L.

(d) Find the eigenvectors and eigenvalues of M.

5. Let A be a matrix with eigenvector v with eigenvalue λ. Show that v is
2
also an eigenvector for A and find the corresponding eigenvalue. How
n
about for A where n ∈ N? Suppose that A is invertible. Show that v
−1
is also an eigenvector for A .
2
6. A projection is a linear operator P such that P = P. Let v be an
eigenvector with eigenvalue λ for a projection P, what are all possible
values of λ? Show that every projection P has at least one eigenvector.
Note that every complex matrix has at least 1 eigenvector, but you
need to prove the above for any field.

7. Explain why the characteristic polynomial of an n × n matrix has de-
gree n. Make your explanation easy to read by starting with some
simple examples, and then use properties of the determinant to give a
general explanation.

8. Compute the characteristic polynomial P M (λ) of the matrix

a b
M = .
c d
Now, since we can evaluate polynomials on square matrices, we can
plug M into its characteristic polynomial and find the matrix P M (M).

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