7.E: Exercises for Chapter 7
( \newcommand{\kernel}{\mathrm{null}\,}\)
Calculational Exercises
1. Let T∈L(F2,F2) be defined by
T(u,v)=(v,u)
for every u,v∈F. Compute the eigenvalues and associated eigenvectors for T.
2. Let T∈L(F3,F3) be defined by
T(u,v,w)=(2v,0,5w)
for every u,v,w∈F. Compute the eigenvalues and associated eigenvectors for T.
3. Let n∈Z+ be a positive integer and T∈L(Fn,Fn) be defined by
T(x1,…,xn)=(x1+⋯+xn,…,x1+⋯+xn)
for every x1,…,xn∈F. Compute the eigenvalues and associated eigenvectors for T.
4. Find eigenvalues and associated eigenvectors for the linear operators on F2 defined by each given 2×2 matrix.
(a)[308−1], (b)[10−94−2], (c)[0340],
(d)[−2−712], (e)[0000], (f)[1001]
Hint: Use the fact that, given a matrix A=[abcd]∈F2×2,λ∈F is an eigenvalue for A if and only if (a−λ)(d−λ)−bc=0.
5. For each matrix A below, find eigenvalues for the induced linear operator T on Fn without performing any calculations. Then describe the eigenvectors v∈Fn associated to each eigenvalue λ by looking at solutions to the matrix equation (A−λI)v=0, where I denotes the identity map on Fn.
(a)[−1605], (b)[−130000−1300001000012], (c)[137110123800040002]
6. For each matrix A below, describe the invariant subspaces for the induced linear operator TonF2 that maps each v∈F2 to T(v)=Av.
(a)[4−121], (b)[01−10], (c)[2302], (d)[1000]
7. Let T∈L(R2) be defined by
T(xy)=(xx+y), for all (xy)∈R2.
Define two real numbers λ+ and λ− as follows:
λ+=1+√52, λ−=1−√52.
(a) Find the matrix of T with respect to the canonical basis for R2 (both as the domain and the codomain of T ; call this matrix A).
(b) Verify that λ+ and λ− are eigenvalues of T by showing that v+ and v− are eigen-
vectors, where
v+=(1λ+), v−=(1λ−).
(c) Show that (v+,v−) is a basis of R2.
(d) Find the matrix of T with respect to the basis (v+,v−) for R2 (both as the domain
and the codomain of T ; call this matrix B).
Proof-Writing Exercises
1. Let V be a finite-dimensional vector space over F with T∈L(V,V), and let U1,…,Um be subspaces of V that are invariant under T. Prove that U1+⋯+Um must then also be an invariant subspace of V under T.
2. Let V be a finite-dimensional vector space over F with T∈L(V,V), and suppose that U1 and U2 are subspaces of V that are invariant under T. Prove that U1∩U2 is also an invariant subspace of V under T.
3. Let V be a finite-dimensional vector space over F with T∈L(V,V) invertible and λ∈F={0}. Prove λ is an eigenvalue for T if and only if λ−1 is an eigenvalue for T−1.
4. Let V be a finite-dimensional vector space over F, and suppose that T∈L(V,V) has the property that every v∈V is an eigenvector for T . Prove that T must then be a scalar multiple of the identity function on V.
5. Let V be a finite-dimensional vector space over F, and let S,T∈L(V) be linear operators on V with S invertible. Given any polynomial p(z)∈F[z], prove that
p(S∘T∘S−1)=S∘p(T)∘S−1.
6. Let V be a finite-dimensional vector space over C,T∈L(V) be a linear operator on V , and p(z)∈C[z] be a polynomial. Prove that λ∈C is an eigenvalue of the linear operator p(T)∈L(V) if and only if T has an eigenvalue μ∈C such that p(μ)=λ.
7. Let V be a finite-dimensional vector space over C with T∈L(V) a linear operator on V. Prove that, for each k=1,…,dim(V), there is an invariant subspace Uk of V under T such that dim(Uk)=k.
8. Prove or give a counterexample to the following claim:
Claim. Let V be a finite-dimensional vector space over F, and let T∈L(V) be a linear operator on V . If the matrix for T with respect to some basis on V has all zeros on the diagonal, then T is not invertible.
9. Prove or give a counterexample to the following claim:
Claim. Let V be a finite-dimensional vector space over F, and let T∈L(V) be a linear
operator on V . If the matrix for T with respect to some basis on V has all non-zero elements on the diagonal, then T is invertible.
10. Let V be a finite-dimensional vector space over F, and let S,T∈L(V) be linear operators on V . Suppose that T has dim(V) distinct eigenvalues and that, given any eigenvector v∈V for T associated to some eigenvalue λ∈F, v is also an eigenvector
for S associated to some (possibly distinct) eigenvalue μ∈F. Prove that T∘S=S∘T.
11. Let V be a finite-dimensional vector space over F, and suppose that the linear operator P∈L(V) has the property that P2=P. Prove that V=null(P)⊕range(P).
12. (a) Let a,b,c,d∈F and consider the system of equations given by
ax1+bx2=0cx1+dx2=0.
Note that x1=x2=0 is a solution for any choice of a,b,c, and d. Prove that this system of equations has a non-trivial solution if and only if ad−bc=0.
(b) Let A=[abcd]∈F2×2 , and recall that we can define a linear operator T∈L(F2) on F2 by setting T(v)=Av for each v=[v1v2]∈F2.
Show that the eigenvalues for T are exactly the λ∈F for which p(λ)=0, where p(z)=(a−z)(d−z)−bc.
Hint: Write the eigenvalue equation Av=λv as (A−λI)v=0 and use the first part.
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.