Page 249 - 35Linear Algebra

13.4 Review Problems 249

n
(t , . . . , , t, 1) for the codomain. Determine if this derivative operator
is diagonalizable.
Recall from Chapter 6 that the derivative operator is linear .

2. When writing a matrix for a linear transformation, we have seen that
the choice of basis matters. In fact, even the order of the basis matters!

(a) Write all possible reorderings of the standard basis (e 1 , e 2 , e 3 )
3
for R .
(b) Write each change of basis matrix between the standard basis
and each of its reorderings. Make as many observations as you
can about these matrices. what are their entries? Do you notice
anything about how many of each type of entry appears in each
row and column? What are their determinants? (Note: These
matrices are known as permutation matrices.)
3
3
(c) Given L : R → R is linear and
   
x 2y − z
y
L   =  3x 
z 2z + x + y

write the matrix M for L in the standard basis, and two reorder-
ings of the standard basis. How are these matrices related?

3. Let
X = {♥, ♣, ♠} , Y = {∗, ?} .

0
0
Write down two different ordered bases, S, S and T, T respectively,
X Y
for each of the vector spaces R and R . Find the change of basis
matrices P and Q that map these bases to one another. Now consider
the map
` : Y → X ,

where `(∗) = ♥ and `(?) = ♠. Show that ` can be used to define a
X Y
linear transformation L : R → R . Compute the matrices M and
0
0
0
M of L in the bases S, T and then S , T . Use your change of basis
−1
0
matrices P and Q to check that M = Q MP.
4. Recall that tr MN = tr NM. Use this fact to show that the trace of a
square matrix M does not depend on the basis you used to compute M.

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