Linear Algebra II
Exercise Sheet no. 1
SS 2011
Prof. Dr. Otto April 11, 2011
Dr. Le Roux Dr. Linshaw
Exercise G1 (Warm-up)
In R3, let g be a line through the origin and E be a plane through the origin such that g is not in E. Determine (geometrically) the eigenvalues and eigenspaces of the following linear maps:
(a) reflection in the planeE.
(b) central reflection in the origin.
(c) parallel projection in the direction ofgontoE.
(d) rotation aboutgthrough 13πfollowed by rescaling in the direction ofgwith factor6.
Which of these maps admit a basis of eigenvectors?
Exercise G2 (Warm-up)
(a) Suppose thatϕ:V →V is a linear map over an arbitrary field, and such that all vectorsv∈Vare eigenvectors of ϕ. Show thatϕmust have exactly one eigenvalueλ, and thatϕis preciselyλ·id, where id is the identity map.
(b) Letψ:R4→R4be the map defined by
ϕ
x y z w
=
x y
−w z
.
Find the (real) eigenvalues ofϕand their multiplicity, and find bases for the corresponding eigenspaces.
Exercise G3 (Fixed points of affine maps)
Recall that an affine map is a functionϕ:R2→R2of the formϕ(x) =ϕ0(x) +bwhereϕ0is a linear map andb∈R2 is a vector. In this exercise we are interested in the question of whether such a mapϕhas afixed point,i.e., a pointx such thatϕ(x) =x.
(a) Prove thatϕhas a fixed point, provided that1is not an eigenvalue ofϕ0.
(b) Letϕbe a rotation through the angleαabout a pointc. Give a formula forϕw.r.t. the standard basis, i.e., find functions f and gsuch thatϕ(x,y) = (f(x,y),g(x,y)).
(c) Let%α:R2→R2be a rotation through the angleα(about the origin) and letτc:x7→x+cbe the translation byc.
Using (ii), show that the compositionτc◦%α◦τ−cis a rotation throughαabout the pointc.
(d) Suppose that the linear mapϕ0is a rotation through an angleα6=0. Prove that the affine mapϕ:x7→ϕ0(x) +b has a fixed pointcand thatϕ=τc◦%α◦τ−c, i.e.,ϕis a rotation throughαaboutc.
(Bonus question: how can you find the centrecgeometrically(i.e., without computation)?)
(e) Give an example of an affine mapϕ(x) =ϕ0(x) +bwithout fixed points such thatϕ0is not the identity map.
Exercise G4 (Eigenvalues and eigenvectors) Consider the real2×2matrixA=
−2 6
−2 5
and the linear mapϕ=ϕAgiven byAw.r.t. the standard basis.
(a) Calculate the eigenvalues ofAby expanding det(A−λE)and find the zeroes/roots of the characteristic polynomial.
(b) For each eigenvalueλidetermine the eigenspaceVλ
i.
(c) Find a basisB ofR2that only consists of eigenvectors ofϕ and find the matrix of the mapϕwith respect to the basisB.
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