Linear Algebra

244 Chapter 3. Maps Between Spaces
and check it against ˆ D
11 + 3 √ 3
6
·
� �
−1
+
0
1+3√3 ·
6
to see that it is the same result as above.
� �
2
=
3
2.2 Example On R3 the map
⎛
⎝ x
⎞
y⎠
z
t
⎛ ⎞
y + z
↦−→ ⎝x + z⎠
x + y
� √
(−3+ 3)/2
(1 + 3 √ �
3)/2
that is represented with respect to the standard basis in this way
⎛
0 1
⎞
1
RepE3,E3 (t) = ⎝1
1
0
1
1⎠
0
can also be represented with respect to another basis
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1
if B = 〈 ⎝−1⎠
, ⎝ 1 ⎠ , ⎝1⎠〉
0 −2 1
⎛
−1
then RepB,B(t) = ⎝ 0
0
0
−1
0
⎞
0
0⎠
2
in a way that is simpler, in that the action of a diagonal matrix is easy to
understand.
Naturally, we usually prefer basis changes that make the representation easier
to understand. When the representation with respect to equal starting and
ending bases is a diagonal matrix we say the map or matrix has been diagonalized.
In Chaper Five we shall see which maps and matrices are diagonalizable,
and where one is not, we shall see how to get a representation that is nearly
diagonal.
We finish this subsection by considering the easier case where representations
are with respect to possibly different starting and ending bases. Recall
that the prior subsection shows that a matrix changes bases if and only if it
is nonsingular. That gives us another version of the above arrow diagram and
equation (∗).
2.3 Definition Same-sized matrices H and ˆ H are matrix equivalent if there
are nonsingular matrices P and Q such that ˆ H = PHQ.
2.4 Corollary Matrix equivalent matrices represent the same map, with respect
to appropriate pairs of bases.
Exercise 19 checks that matrix equivalence is an equivalence relation. Thus
it partitions the set of matrices into matrix equivalence classes.