1 Linear Algebra Professor M. Zuker
Problem Set 6
Chapters 4, 5 and extra.
Note: Z1, Z2 etc. refer to my own problems, which may be very close to
what is in the text.
Z1. This is problem 2 from Chapter 4. Suppose that z1, . . . , zm+1 are distinct
elements of Fand w1, . . . , wm+1 โF. Prove that there exists a unique poly-
nomial pโ Pm(F) such that
p(zj) = wj
for j= 1, . . . , m + 1.
Hint: Define a linear map from Pm(F) to Fm+1.
Z2. Suppose that Pโ L(V) satisfies P2=P. Compute all possible eigenvalues
of P.
22. Suppose V=UโW, where Uand Ware nonzero subspaces of V. Find all
eigenvalues and eigenvectors of PU,W .
Comment: You may wish to delay this problem until I discuss this
particular operator in class on Tuesday.
Z3. Suppose that Tโ L(R2) has matrix
๎aโb
b a ๎,
where a, b โRand b6= 0. Prove that Thas no eigenvalues.
23. Give an example of an operator Tโ L(R4) such that Thas no (real) eigen-
values. Can you generalize this to Tโ L(R2n) for any positive integer, n?
