MATH 731, FALL 2008
HOMEWORK SET 1
Originally due Wednesday, September 10 – due date changed to Friday, September 12
A. Let A, B be arbitrary sets and let f:A→Bbe a function. Prove that the following
statements are equivalent.
(1) fis onto.
(2) fis right invertible, that is, there exists g:B→Awith f◦g= idB.
(3) fis right cancellable, that is, if g , h :B→Care any functions with g◦f=h◦f,
then g=h.
B. (i) Show that a group Gis Abelian if and only if the map φ:G→Gdefined by
φ(g) = g−1is a group homomorphism.
(ii) Determine for which fields Fthe map φ:F→Fdefined by φ(0) = 0 and φ(x) = x−1
for x6= 0 is a field homomorphism.
C. Let Vbe a vector space and let π:V→Vbe a linear operator. We say πis an internal
projection if π2=π. (The word “internal” is often omitted.)
(i) Let π:V→Vbe an internal projection. Prove that V= im π⊕ker π.
(ii) Suppose W, W 0are subspaces of Vwith V=W⊕W0. Prove that there is an internal
projection π:V→Vwith im π=W,π|W= idW, and ker π=W0.
D. Let Vbe a vector space over the field Fand let U, W be subspaces. Prove that
dim(U+W) + dim(U∩W) = dim U+ dim W.
E. Find all 2 ×2 matrices Aover Rsatisfying A2=−I.
Hints: There are many such matrices. First show that x2+ 1 must be both the minimal
and characteristic polynomial of A. The “characteristic” fact gives you the trace and
determinant of A.
Notes: B(ii) is probably hard - it depends on some knowledge of fields.
For C, recall that V=W⊕W0means V=W+W0and W∩W0={0}.
D can be done by using the Dimension Formula and the First Isomorphism Theorem.