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7.E: Exercises for Chapter 7

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Calculational Exercises

1. Let TL(F2,F2) be defined by

T(u,v)=(v,u)


for every u,vF. Compute the eigenvalues and associated eigenvectors for T.

2. Let TL(F3,F3) be defined by

T(u,v,w)=(2v,0,5w)

for every u,v,wF. Compute the eigenvalues and associated eigenvectors for T.

3. Let nZ+ be a positive integer and TL(Fn,Fn) be defined by

T(x1,,xn)=(x1++xn,,x1++xn)

for every x1,,xnF. 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)[3081],  (b)[10942],  (c)[0340],

(d)[2712],  (e)[0000],  (f)[1001]


Hint: Use the fact that, given a matrix A=[abcd]F2×2,λF is an eigenvalue for A if and only if (aλ)(dλ)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 vFn associated to each eigenvalue λ by looking at solutions to the matrix equation (AλI)v=0, where I denotes the identity map on Fn.

(a)[1605],  (b)[1300001300001000012],  (c)[137110123800040002]

6. For each matrix A below, describe the invariant subspaces for the induced linear operator TonF2 that maps each vF2 to T(v)=Av.

(a)[4121],  (b)[0110],  (c)[2302],  (d)[1000]

7. Let TL(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=152.


(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 TL(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 TL(V,V), and suppose that U1 and U2 are subspaces of V that are invariant under T. Prove that U1U2 is also an invariant subspace of V under T.

3. Let V be a finite-dimensional vector space over F with TL(V,V) invertible and λF={0}. Prove λ is an eigenvalue for T if and only if λ1 is an eigenvalue for T1.

4. Let V be a finite-dimensional vector space over F, and suppose that TL(V,V) has the property that every vV 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,TL(V) be linear operators on V with S invertible. Given any polynomial p(z)F[z], prove that

p(STS1)=Sp(T)S1.

6. Let V be a finite-dimensional vector space over C,TL(V) be a linear operator on V , and p(z)C[z] be a polynomial. Prove that λC is an eigenvalue of the linear operator p(T)L(V) if and only if T has an eigenvalue μC such that p(μ)=λ.

7. Let V be a finite-dimensional vector space over C with TL(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 TL(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 TL(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,TL(V) be linear operators on V . Suppose that T has dim(V) distinct eigenvalues and that, given any eigenvector vV for T associated to some eigenvalue λF, v is also an eigenvector
for S associated to some (possibly distinct) eigenvalue μF. Prove that TS=ST.

11. Let V be a finite-dimensional vector space over F, and suppose that the linear operator PL(V) has the property that P2=P. Prove that V=null(P)range(P).

12. (a) Let a,b,c,dF 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 adbc=0.

(b) Let A=[abcd]F2×2 , and recall that we can define a linear operator TL(F2) on F2 by setting T(v)=Av for each v=[v1v2]F2.

Show that the eigenvalues for T are exactly the λF for which p(λ)=0, where p(z)=(az)(dz)bc.

Hint: Write the eigenvalue equation Av=λv as (AλI)v=0 and use the first part.


This page titled 7.E: Exercises for Chapter 7 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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