Algebra Qual 2019-01

Problem 1

Let $W\subset {\mathbf{R}}^5$ be the subspace spanned by the set of vectors $\{⟨1,-2,0,2,-1⟩,⟨-2,4,-1,1,2⟩,⟨0,1,2,-2,1⟩\}$.
(a) Compute the dimension of $W$.
(b) Determine the dimension of $W^{\perp }$, the perpendicular subspace in ${\mathbf{R}}^5$.
(c) Find a basis for $W^{\perp }$.

Problem 2

Let $D$ be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.

Problem 3

Let $G$ be a group and suppose $\mathrm{Aut}(G)$ is trivial.
(a) Show that $G$ is abelian.
(b) Show that for any abelian group $H$, the inversion map $\varphi (h)=h^{-1}$ is an automorphism.
(c) Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.

Problem 4

Suppose $H$ is a group of order 15. Prove there does not exist a nontrivial group homomorphism $\varphi :D_5\to H$, where $D_5$ is the dihedral group with ten elements.

Problem 5

Let $a,b\in \mathbf{R}$ and $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the linear transformation that is orthogonal projection onto the plane $z=ax+by$ (with respect to the usual Euclidean inner-product on ${\mathbf{R}}^3$).
(a) Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
(b) Is $T$ diagonalizable? Justify.
(c) What is the characteristic polynomial of $T$?