Problem Set 9 (due November 12)
MATH 110: Linear Algebra
Each problem is worth 10 points.
PART 1
1. Curtis p. 161 10.
2. Curtis p. 226 7.
3. Curtis p. 243 6.
PART 2
Problem 1(20)
Let Vbe a real vector space of functions spanned by the set of real
values functions {ex, xex, x2ex, e2x}and let Tbe the linear transformation
T:VโVdefined by T(f) = f0, the derivative of f. Find the Jordan
canonical form of T.
Problem 2(10)
Prove that if Vis isomorphic to Wthen Vโis isomorphic to Wโ. Is the
converse true (prove or give a counterexample)?
Problem 3 (10)
a) Let T:RโRbe a linear transformation. Show that T(x) = cx where
cโRis some constant.
b) Let T:R2โRbe a linear transformation. Show that T(x, y) =
c1x+c2y.
c) Generalize parts a) and b) to a linear transformation of the form T:
RnโR.
d) Show that every plane through the origin in R3may be identified with
the null space of an element in (R3)โ.
Problem 4 (10)
Let T:Mn,n(R)โMn,n(R) be a linear transformation from the vector
space of nรnmatrices over Rinto itself, where T(A) = At. Find the minimal
polynomial of T.
PART 3 - Optional Problem
Prove the lights and switches result combinatorially (i.e., without using
linear algebra).
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