LECTURE EXERCISES #1
AARON BERTRAM, PCMI SUMMER SCHOOL 2015
1) Recall the categorical definition of a Cartesian product in the category of SET:
Given sets S and T , the Cartesian product is a set P along with (set) maps
πS : U → S and πT : U → T
such that, given any set X and maps
fS : X → S and fT : X → T
there is a unique map f : X → P such that the following diagram commutes:
S
fS
X
∃!f
πS
fT
T
πT
P
Show that setting P = S ×T (along with a suitable choice of πS and πT ) satisfies
this requirement. Pay attention to the uniqueness condition!
2) Calculate the eigenvalues and eigenvectors of
cos θ
sin θ
cos θ
and
sin θ − cos θ
sin θ
− sin θ
.
cos θ
Note in each of the two cases where there is a dependence on θ.
3) An n × n matrix A is said to be diagonalizable if there exists an invertible
n × n matrix M and a diagonal matrix D such that
M AM −1 = D.
Show that if A is diagonalizable, then the columns of M are the eigenvectors of A,
and the entries of D are the corresponding eigenvalues.
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