Problem solving seminar I
1. Let T be a linear transformation of a vector space V into itself. Suppose that
v ∈ V is such that Tm
v = 0, Tm−1
v = 0 for some positive integer m. Show that
v, Tv, T2
v, . . . , Tm−1
v are linearly independent.
2. Let A be an n × n matrix over a field K. Prove that
rank A2
− rank A3
≤ rank A − rank A2
.
3. (a) Prove that there is no continuous function from the closed interval [0, 1] onto
the open interval (0, 1).
(b) Find a continuous surjective function from the open interval (0, 1) onto the closed
interval [0, 1].
(c) Prove that no map from (b) is bijective.
4. Compute the 100th derivation of the function
x2
+ 1
x3 − x
.
5. Suppose that f is a continuous real function with period 1. Show that there is a
real number x0 such that
f(x0 + π) = f(x0).
Homework I. Find the limit
lim
x→0
sin tan x − tan sin x
arcsin arctan x − arctan arcsin x
.
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