1. Label the following statements as true or false.

   1. (a) There exists a linear operator T with no T-invariant subspace.

   2. (b) If T is a linear operator on a finite-dimensional vector space V and W is a T-invariant subspace of V, then the characteristic polynomial of ${\text{T}}_{\text{W}}$ divides the characteristic polynomial of T.

   3. (c) Let T be a linear operator on a finite-dimensional vector space V, and let v and w be in V. If W is the T-cyclic subspace generated by v, ${\text{W}}^{\prime }$ is the T-cyclic subspace generated by w, and $\text{W}={\text{W}}^{\prime }$, then $v=w$.

   4. (d) If T is a linear operator on a finite-dimensional vector space V, then for any $v\in \text{V}$ the T-cyclic subspace generated by v is the same as the T-cyclic subspace generated by T(v).

   5. (e) Let T be a linear operator on an n-dimensional vector space. Then there exists a polynomial g(t) of degree n such that $g(\text{T})={\text{T}}_0$.

   6. (f) Any polynomial of degree n with leading coefficient ${(-1)}^n$ is the characteristic polynomial of some linear operator.

   7. (g) If T is a linear operator on a finite-dimensional vector space V, and if V is the direct sum of k T-invariant subspaces, then there is an ordered basis $\beta$ for V such that ${[\text{T}]}_{\beta }$ is a direct sum of k matrices.

2. For each of the following linear operators T on the vector space V, determine whether the given subspace W is a T-invariant subspace of V.

   1. (a) $\text{V}={\text{P}}_3(R),\text{T}(f(x))=f^{\prime }(x)$, and $\text{W}={\text{P}}_2(R)$

   2. (b) $\text{V}=\text{P}(R),\text{T}(f(x))=xf(x)$, and $\text{W}={\text{P}}_2(R)$

   3. (c) $\text{V}={\text{R}}^3,\text{T}(a,b,c)=(a+b+c,a+b+c,a+b+c)$, and $\text{W}=\{(t,t,t):t\in R\}$

   4. (d) $\text{V}=\text{C}([0,1]),\text{T}(f(t))=\left[{\displaystyle {\int }_0^{1}f(x)dx}\right]t$, and $\text{W}=\{f\in \text{V}:f(t)=at+b\text{ for some }a\text{ and }b\}$

   5. (f) $\text{V}={\text{M}}_{2\times 2}(R),\text{T(}A\text{)=}\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right)A$, and $\text{W}=\{A\in \text{V}:A^t=A\}$

3. Let T be a linear operator on a finite-dimensional vector space V. Prove that the following subspaces are T-invariant.

   1. (a) $\left\{0\right\}$ and V

   2. (b) N(T) and R(T)

   3. (c) ${\text{E}}_{\lambda }$, for any eigenvalue $\lambda$ of T

4. Let T be a linear operator on a vector space V, and let W be a T-invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t).

5. Let T be a linear operator on a vector space V. Prove that the intersection of any collection of T-invariant subspaces of V is a T-invariant subspace of V.

6. For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z.

   1. (a) $\text{V}={\text{R}}^4,\text{T}(a,b,c,d)=(a+b,b-c,a+c,a+d)$, and $z=e_1$.

   2. (b) $\text{V}={\text{P}}_3(R),\text{T}(f(x))=f^{″}(x)$, and $z=x^3$.

   3. (c) $\text{V}={\text{M}}_{2\times 2}(R),\text{T}(A)=A^t$, and $z=\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right)$.

   4. (d) $\text{V}={\text{M}}_{2\times 2}(R),\text{T}(A)=\left(\begin{array}{rr}1& 1\\ 2& 2\end{array}\right)A$, and $z=\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right)$.

7. Prove that the restriction of a linear operator T to a T-invariant sub-space is a linear operator on that subspace.

8. Let T be a linear operator on a vector space with a T-invariant subspace W. Prove that if v is an eigenvector of ${\text{T}}_{\text{W}}$ with corresponding eigenvalue $\lambda$, then v is also an eigenvector of T with corresponding eigenvalue $\lambda$.

9. For each linear operator T and cyclic subspace W in Exercise 6, compute the characteristic polynomial of ${\text{T}}_{\text{W}}$ in two ways, as in Example 6.

10. For each linear operator in Exercise 6, find the characteristic polynomialf(t) of T, and verify that the characteristic polynomial of ${\text{T}}_{\text{W}}$ (computed in Exercise 9) dividesf(t).

11. Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. Prove that

    1. (a) W is T-invariant.

    2. (b) Any T-invariant subspace of V containing v also contains W.

12. Prove that $A=\left(\begin{array}{rr}{B}_1& B_2\\ \hfill O\hfill & B_3\end{array}\right)$ in the proof of Theorem 5.20.

13. Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. For any $w\in \text{V}$, prove that $w\in \text{W}$ if and only if there exists a polynomial g(t) such that $w=g(\text{T})(v)$.

14. Prove that the polynomial g(t) of Exercise 13 can always be chosen so that its degree is less than $\text{dim}(\text{W})$.

15. Use the Cayley-Hamilton theorem (Theorem 5.22) to prove its corollary for matrices. Warning: If $f(t)=\text{det}(A-tI)$ is the characteristic polynomial of A, it is tempting to “prove” that $f(A)=O$ by saying “$f(A)=\text{det}(A-AI)=\text{det}(O)=0$.” Why is this argument incorrect? Visit goo.gl/ZMVn9i for a solution.

16. Let T be a linear operator on a finite-dimensional vector space V.

    1. (a) Prove that if the characteristic polynomial of T splits, then so does the characteristic polynomial of the restriction of T to any T-invariant subspace of V.

    2. (b) Deduce that if the characteristic polynomial of T splits, then any nontrivial T-invariant subspace of V contains an eigenvector of T.

17. Let A be an $n\times n$ matrix. Prove that

    $$\text{dim}(\text{span}(\{I_n,A,A^2,\dots \}))\le n.$$

18. Let A be an $n\times n$ matrix with characteristic polynomial

    $$f(t)={(-1)}^n{t}^n+a_{n-1}{t}^{n-1}+\cdots +a_{1}t+a_0.$$

    1. (a) Prove that A is invertible if and only if $a_0\ne 0$.

    2. (b) Prove that if A is invertible, then

       $$A^{-1}=(-1/a_0)[{(-1)}^n{A}^{n-1}+a_{n-1}{A}^{n-2}+\cdots +a_1{I}_n].$$

    3. (c) Use (b) to compute $A^{-1}$ for

       $$A=\left(\begin{array}{rrr}1& 2& 1\\ 0& 2& 3\\ 0& 0& -1\end{array}\right).$$

19. Let A denote the $k\times k$ matrix

    $$\left(\begin{array}{ccccc}0& 0& \cdots & 0& -a_0\\ 1& 0& \cdots & 0& -a_1\\ 0& 1& \cdots & 0& -a_2\\ ⋮& ⋮& & ⋮& ⋮\\ 0& 0& \cdots & 1& -a_{k-1}\end{array}\right),$$

    where $a_0,a_1,\dots ,a_{k-1}$ are arbitrary scalars. Prove that the characteristic polynomial of A is

    $${(-1)}^k(a_0+a_{1}t+\cdots +a_{k-1}{t}^{k-1}+t^k).$$

    Hint: Use mathematical induction on k, computing the determinant by cofactor expansion along the first row.

20. Let T be a linear operator on a vector space V, and suppose that V is a T-cyclic subspace of itself. Prove that if U is a linear operator on V, then $\text{UT}=\text{TU}$ if and only if $\text{U}=g(\text{T})$ for some polynomial g(t). Hint: Suppose that V is generated by v. Choose g(t) according to Exercise 13 so that $g(\text{T})(v)=\text{U}(v)$.

21. Let T be a linear operator on a two-dimensional vector space V. Prove that either V is a T-cyclic subspace of itself or $\text{T}=c\text{I}$ for some scalar c.

22. Let T be a linear operator on a two-dimensional vector space V and suppose that $\text{T}\ne c\text{I}$ for any scalar c. Show that if U is any linear operator on V such that $\text{UT}=\text{TU}$, then $\text{U}=g(\text{T})$ for some polynomial g(t).

23. Let T be a linear operator on a finite-dimensional vector space V, and let W be a T-invariant subspace of V. Suppose that $v_1,v_2,\dots ,v_k$ are eigenvectors of T corresponding to distinct eigenvalues. Prove that if $v_1+v_2+\cdots +v_k$ is in W, then $v_i\in \text{W}$ for all i. Hint: Use mathematical induction on k.

24. Prove that the restriction of a diagonalizable linear operator T to any nontrivial T-invariant subspace is also diagonalizable. Hint: Use the result of Exercise 23.

25. 1. (a) Prove the converse to Exercise 19(a) of Section 5.2: If T and U are diagonalizable linear operators on a finite-dimensional vector space V such that $\text{UT}=\text{TU}$, then T and U are simultaneously diagonalizable. (See the definitions in the exercises of Section 5.2.) Hint: For any eigenvalue $\lambda$ of T, show that ${\text{E}}_{\lambda }$ is U-invariant, and apply Exercise 24 to obtain a basis for ${\text{E}}_{\lambda }$ of eigenvectors of U.

    2. (b) State and prove a matrix version of (a).

26. Let T be a linear operator on an n-dimensional vector space V such that T has n distinct eigenvalues. Prove that V is a T-cyclic subspace of itself. Hint: Use Exercise 23 to find a vector v such that $\{v,\text{T}(v),\dots ,{\text{T}}^{n-1}(v)\}$ is linearly independent.

Exercises 27 through 31 require familiarity with quotient spaces as defined in Exercise 31 of Section 1.3. Before attempting these exercises, the reader should first review the other exercises treating quotient spaces: Exercise 35 of Section 1.6, Exercise 42 of Section 2.1, and Exercise 24 of Section 2.4.

For the purposes of Exercises 27 through 31, T is a fixed linear operator on a finite-dimensional vector space V, and W is a nonzero T-invariant subspace of V. We require the following definition.

27. 1. (a) Prove that $\overline{\text{T}}$ is well defined. That is, show that $\overline{\text{T}}(v+\text{W})=\overline{\text{T}}(v^{\prime }+\text{W})$ whenever $v+\text{W}=v^{\prime }+\text{W}$.

    2. (b) Prove that $\overline{\text{T}}$ is a linear operator on V/W.

    3. (c) Let $\eta :\text{V}\to \text{V}/\text{W}$ be the linear transformation defined in Exercise 42 of Section 2.1 by $\eta (v)=v+\text{W}$. Show that the diagram of Figure 5.6 commutes; that is, prove that $\eta \text{T}=\overline{\text{T}}\eta$. (This exercise does not require the assumption that V is finite-dimensional.)

    

    Figure 5.6 Full Alternative Text

28. Letf(t), g(t), and h(t) be the characteristic polynomials of T, ${\text{T}}_{\text{W}}$, and $\overline{\text{T}}$, respectively. Prove that $f(t)=g(t)h(t)$. Hint: Extend an ordered basis $\gamma =\{v_1,v_2,\dots ,v_k\}$ for W to an ordered basis $\beta =\{v_1,v_2,\dots ,v_k,v_{k+1},\dots ,v_n\}$ for V. Then show that the collection of cosets $\alpha =\{v_{k+1}+\text{W},v_{k+2}+\text{W},\dots ,v_n+\text{W}\}$ is an ordered basis for V/W, and prove that

    $${[\text{T}]}_{\beta }=\left(\begin{array}{rr}{B}_1& B_2\\ \hfill O\hfill & B_3\end{array}\right),$$

    where $B_1={[\text{T}]}_{\gamma }$ and $B_3={[\overline{\text{T}}]}_{\alpha }$.

29. Use the hint in Exercise 28 to prove that if T is diagonalizable, then so is $\overline{\text{T}}$.

30. Prove that if both ${\text{T}}_{\text{W}}$ and $\overline{\text{T}}$ are diagonalizable and have no common eigenvalues, then T is diagonalizable.

The results of Theorem 5.21 and Exercise 28 are useful in devising methods for computing characteristic polynomials without the use of determinants. This is illustrated in the next exercise.

31. Let $A=\left(\begin{array}{rrr}1& 1& -3\\ 2& 3& 4\\ 1& 2& 1\end{array}\right)$, let $\text{T}={\text{L}}_A$, and let W be the cyclic subspace of ${\text{R}}^3$ generated by $e_1$.

    1. (a) Use Theorem 5.21 to compute the characteristic polynomial of ${\text{T}}_{\text{W}}$.

    2. (b) Show that $\{e_2+\text{W}\}$ is a basis for ${\text{R}}^3/\text{W}$, and use this fact to compute the characteristic polynomial of $\overline{\text{T}}$.

    3. (c) Use the results of (a) and (b) to find the characteristic polynomial of A.

Exercises 32 through 39 are concerned with direct sums.

32. Let T be a linear operator on a vector space V, and let ${\text{W}}_1,{\text{W}}_2,\dots {\text{W}}_k$ be T-invariant subspaces of V. Prove that ${\text{W}}_1+{\text{W}}_2+\cdots +{\text{W}}_k$ is also a T-invariant subspace of V.

33. Give a direct proof of Theorem 5.24 for the case $k=2$. (This result is used in the proof of Theorem 5.23.)

34. Prove Theorem 5.24. Hint: Begin with Exercise 33 and extend it using mathematical induction on k, the number of subspaces.

35. Let T be a linear operator on a finite-dimensional vector space V. Prove that T is diagonalizable if and only if V is the direct sum of one-dimensional T-invariant subspaces.

36. Let T be a linear operator on a finite-dimensional vector space V, and let ${\text{W}}_1,{\text{W}}_2,\dots {\text{W}}_k$ be T-invariant subspaces of V such that $\text{V}={\text{W}}_1\oplus {\text{W}}_2\oplus \cdots \oplus {\text{W}}_k$. Prove that

    $$\text{det}(\text{T})=\text{det}({\text{T}}_{{\text{W}}_1})·\text{det}({\text{T}}_{{\text{W}}_2})·\cdots ·\text{det}({\text{T}}_{{\text{W}}_k}).$$

37. Let T be a linear operator on a finite-dimensional vector space V, and let ${\text{W}}_1,{\text{W}}_2,\dots ,{\text{W}}_k$ be T-invariant subspaces of V such that $\text{V}={\text{W}}_1\oplus {\text{W}}_2\oplus \cdots \oplus {\text{W}}_k$. Prove that T is diagonalizable if and only if ${\text{T}}_{{\text{W}}_i}$ is diagonalizable for all i.

38. Let C be a collection of diagonalizable linear operators on a finite- dimensional vector space V. Prove that there is an ordered basis $\beta$ such that ${[\text{T}]}_{{}^{\beta }}$ is a diagonal matrix for all $\text{T}\in C$ if and only if the operators of C commute under composition. (This is an extension of Exercise 25.) Hints for the case that the operators commute: The result is trivial if each operator has only one eigenvalue. Otherwise, establish the general result by mathematical induction on dim(V), using the fact that V is the direct sum of the eigenspaces of some operator in C that has more than one eigenvalue.

39. Let $B_1,B_2,\dots ,B_k$ be square matrices with entries in the same field, and let $A=B_1\oplus B_2\oplus \cdots \oplus B_k$. Prove that the characteristic polynomial of A is the product of the characteristic polynomials of the $B_i\text{'}s$.

40. Let

    $$A=\left(\begin{array}{cccc}1& 2& \cdots & n\\ n+1& n+2& \cdots & 2n\\ ⋮& ⋮& & ⋮\\ n^2-n+1& n^2-n+2& \cdots & n^2\end{array}\right).$$

    Find the characteristic polynomial of A. Hint: First prove that A has rank 2 and that $\text{span}(\{(1,1,\dots ,1),(1,2,\dots ,n)\})$ is ${\text{L}}_A$-invariant.

41. Let $A\in {\text{M}}_{n\times n}(R)$ be the matrix defined by $A_{ij}=1$ for all i and j. Find the characteristic polynomial of A.