9.3E: Invariant Subspaces and Direct Sums Exercises

Exercise \(\PageIndex{1}\)

If \(T : V \to V\) is any linear operator, show that \(\text{ker }T\) and \(im \;T\) are \(T\)-invariant subspaces.

Exercise \(\PageIndex{2}\)

Let \(T\) be a linear operator on \(V\). If \(U\) and \(W\) are \(T\)-invariant, show that

1. \(U \cap W\) and \(U + W\) are also \(T\)-invariant.
2. \(T(U)\) is \(T\)-invariant.

Answer

2. \(T(U) \subseteq U\), so \(T[T(U)] \subseteq T(U)\).

Exercise \(\PageIndex{3}\)

Let \(S\) and \(T\) be linear operators on \(V\) and assume that \(ST = TS\).

1. Show that \(im \;S\) and \(\text{ker }S\) are \(T\)-invariant.
2. If \(U\) is \(T\)-invariant, show that \(S(U)\) is \(T\)-invariant.

Answer

2. If \(\mathbf{v}\) is in \(S(U)\), write \(\mathbf{v} = S(\mathbf{u})\), \(\mathbf{u}\) in \(U\). Then \(T(\mathbf{v}) = T[S(\mathbf{u})] = (TS)(\mathbf{u}) = (ST)(\mathbf{u}) = S[T(\mathbf{u})]\) and this lies in \(S(U)\) because \(T(\mathbf{u})\) lies in \(U\) (\(U\) is \(T\)-invariant).

Exercise \(\PageIndex{4}\)

Let \(T : V \to V\) be a linear operator. Given \(\mathbf{v}\) in \(V\), let \(U\) denote the set of vectors in \(V\) that lie in every \(T\)-invariant subspace that contains \(\mathbf{v}\).

1. Show that \(U\) is a \(T\)-invariant subspace of \(V\) containing \(\mathbf{v}\).
2. Show that \(U\) is contained in every \(T\)-invariant subspace of \(V\) that contains \(\mathbf{v}\).

Exercise \(\PageIndex{5}\)

1. If \(T\) is a scalar operator (see Example 7.1.1) show that every subspace is \(T\)-invariant.
2. Conversely, if every subspace is \(T\)-invariant, show that \(T\) is scalar.

Exercise \(\PageIndex{6}\)

Show that the only subspaces of \(V\) that are \(T\)-invariant for every operator \(T : V \to V\) are \(0\) and \(V\). Assume that \(V\) is finite dimensional. [Hint: Theorem 7.1.3.]

Answer

Suppose \(U\) is \(T\)-invariant for every \(T\). If \(U \neq 0\), choose \(\mathbf{u} \neq \mathbf{0}\) in \(U\). Choose a basis \(B = \{\mathbf{u}, \mathbf{u}_2, \dots, \mathbf{u}_n\}\) of \(V\) containing \(\mathbf{u}\). Given any \(\mathbf{v}\) in \(V\), there is (by Theorem 7.1.3) a linear transformation \(T : V \to V\) such that \(T(\mathbf{u}) = \mathbf{v}\), \(T(\mathbf{u}_2) = \cdots = T(\mathbf{u}_n) = \mathbf{0}\). Then \(\mathbf{v} = T(\mathbf{u})\) lies in \(U\) because \(U\) is \(T\)-invariant. This shows that \(V = U\).

Exercise \(\PageIndex{7}\)

Suppose that \(T : V \to V\) is a linear operator and that \(U\) is a \(T\)-invariant subspace of \(V\). If \(S\) is an invertible operator, put \(T^{\prime} = STS^{-1}\). Show that \(S(U)\) is a \(T^{\prime}\)-invariant subspace.

Exercise \(\PageIndex{8}\)

In each case, show that \(U\) is \(T\)-invariant, use it to find a block upper triangular matrix for \(T\), and use that to compute \(c_{T}(x)\).

1. \(T :\|{P}_2 \to\|{P}_2\), \(T(a + bx + cx^2) = (-a + 2b + c) + (a + 3b +c)x + (a + 4b)x^2\), \(U = span \;\{1, x + x^2\}\)
2. \(T :\|{P}_2 \to\|{P}_2\), \(T(a + bx + cx^2) = (5a - 2b + c) + (5a - b + c)x + (a + 2c)x^2\), \(U = span \;\{1 - 2x^2, x + x^2\}\)

Answer

2. \(T(1 - 2x^{2}) = 3 + 3x - 3x^{2} = 3(1 - 2x^{2}) + 3(x + x^{2})\) and \(T(x + x^{2}) = -(1 - 2x^{2})\), so both are in \(U\). Hence \(U\) is \(T\)-invariant by Example 9.3.3. If \(B = \{ 1 - 2x^2, x + x^2, x^2 \}\) then \(M_B(T) = \left[ \begin{array}{rrr} 3 & -1 & 1 \\ 3 & 0 & 1 \\ 0 & 0 & 3 \end{array} \right]\), so \(c_T(x) = \det \left[ \begin{array}{ccc} x - 3 & 1 & -1 \\ -3 & x & -1 \\ 0 & 0 & x - 3 \end{array} \right] = (x - 3) \det \left[ \begin{array}{cc} x - 3 & 1 \\ -3 & x \end{array} \right] = (x - 3)(x^2 -3x + 3)\)

Exercise \(\PageIndex{9}\)

In each case, show that \(T_{A} : \mathbb{R}^2 \to \mathbb{R}^2\) has no invariant subspaces except \(0\) and \(\mathbb{R}^2\).

1. \(A = \left[ \begin{array}{rr} 1 & 2 \\ -1 & -1 \end{array} \right]\)
2. \(A = \left[ \begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right]\), \(0 < \theta < \pi\)

Answer

2. Suppose \(\mathbb{R}\mathbf{u}\) is \(T_{A}\)-invariant where \(\mathbf{u} \neq \mathbf{0}\). Then \(T_{A}(\mathbf{u}) = r\mathbf{u}\) for some \(r\) in \(\mathbb{R}\), so \((rI - A)\mathbf{u} = \mathbf{0}\). But \(\det (rI - A) = (r - \cos \theta)^{2} + sin^{2} \theta \neq 0\) because \(0 < \theta < \pi\). Hence \(\mathbf{u} = \mathbf{0}\), a contradiction.

Exercise \(\PageIndex{10}\)

In each case, show that \(V = U \oplus W\).

1. \(V = \mathbb{R}^4\), \(U = span \;\{(1, 1, 0, 0), (0, 1, 1, 0)\}\), \(W = span \;\{(0, 1, 0, 1), (0, 0, 1, 1)\}\)
2. \(V = \mathbb{R}^4\), \(U = \{(a, a, b, b) \mid a, b \mbox{ in }\mathbb{R}\}\), \(W = \{(c, d, c, -d) \mid c, d \mbox{ in } \mathbb{R}\}\)
3. \(V =\|{P}_{3}\), \(U = \{a + bx \mid a, b \mbox{ in }\mathbb{R}\}\), \(W = \{ax^{2} + bx^{3} \mid a, b \mbox{ in } \mathbb{R}\}\)
4. \(V =\|{M}_{22}\), \(U = \left\{ \left[ \begin{array}{rr} a & a \\ b & b \end{array} \right] \middle| a, b \mbox{ in } \mathbb{R} \right\}\), \(W = \left\{ \left[ \begin{array}{rr} a & b \\ -a & b \end{array} \right] \middle| a, b \mbox{ in } \mathbb{R} \right\}\)

Answer

2. \(U = span \;\{(1, 1, 0, 0), (0, 0, 1, 1)\}\) and \(W = span \;\{(1, 0, 1, 0), (0, 1, 0, -1)\}\), and these four vectors form a basis of \(\mathbb{R}^4\). Use Example 9.3.9.
4. \(U = span \; \left\{ \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right], \left[ \begin{array}{rr} 0 & 0 \\ 1 & 1 \end{array} \right] \right\}\) and \(W = span \; \left\{ \left[ \begin{array}{rr} 1 & 0 \\ -1 & 0 \end{array} \right], \left[ \begin{array}{rr} 0 & 1 \\ 0 & 1 \end{array} \right] \right\}\) and these vectors are a basis of \(\mathbf{M}_{22}\). Use Example 9.3.9.

Exercise \(\PageIndex{11}\)

Let \(U = span \;\{(1, 0, 0, 0), (0, 1, 0, 0)\}\) in \(\mathbb{R}^4\). Show that \(\mathbb{R}^4 = U \oplus W_{1}\) and \(\mathbb{R}^4 = U \oplus W_{2}\), where \(W_{1} = span \;\{(0, 0, 1, 0), (0, 0, 0, 1)\}\) and \(W_{2} = span \;\{(1, 1, 1, 1), (1, 1, 1, -1)\}\).

Exercise \(\PageIndex{12}\)

Let \(U\) be a subspace of \(V\), and suppose that \(V = U \oplus W_{1}\) and \(V = U \oplus W_{2}\) hold for subspaces \(W_{1}\) and \(W_{2}\). Show that \(dim \;W_{1} = dim \;W_{2}\).

Exercise \(\PageIndex{13}\)

If \(U\) and \(W\) denote the subspaces of even and odd polynomials in \(\mathbf{P}_{n}\), respectively, show that \(\mathbf{P}_{n} = U \oplus W\). (See Exercise 6.3.36.) [Hint: \(f(x) + f(-x)\) is even.]

Exercise \(\PageIndex{14}\)

Let \(E\) be an \(n \times n\) matrix with \(E^{2} = E\). Show that \(\mathbf{M}_{nn} = U \oplus W\), where \(U = \{A \mid AE = A\}\) and \(W = \{B \mid BE = 0\}\). [Hint: \(XE\) lies in \(U\) for every matrix \(X\).]

Answer

The fact that \(U\) and \(W\) are subspaces is easily verified using the subspace test. If \(A\) lies in \(U \cap V\), then \(A = AE = 0\); that is, \(U \cap V = 0\). To show that \(\mathbf{M}_{22} = U + V\), choose any \(A\) in \(\mathbf{M}_{22}\). Then \(A = AE + (A - AE)\), and \(AE\) lies in \(U\) [because \((AE)E = AE^{2} = AE\)], and \(A - AE\) lies in \(W\) [because \((A - AE)E = AE - AE^{2} = 0\)].

Exercise \(\PageIndex{15}\)

Let \(U\) and \(W\) be subspaces of \(V\). Show that \(U \cap W = \{\mathbf{0}\}\) if and only if \(\{\mathbf{u}, \mathbf{w}\}\) is independent for all \(\mathbf{u} \neq \mathbf{0}\) in \(U\) and all \(\mathbf{w} \neq \mathbf{0}\) in \(W\).

Exercise \(\PageIndex{16}\)

Let \(V \stackrel{T}{\to} W \stackrel{S}{\to} V\) be linear transformations, and assume that \(dim \;V\) and \(dim \;W\) are finite.

1. If \(ST = 1_{V}\), show that \(W = im \;T \oplus \text{ker }S\).
2. Illustrate with \(\mathbb{R}^2 \stackrel{T}{\to} \mathbb{R}^3 \stackrel{S}{\to} \mathbb{R}^2\) where
   \(T(x, y) = (x, y, 0)\) and \(S(x, y, z) = (x, y)\).

Exercise \(\PageIndex{17}\)

Let \(U\) and \(W\) be subspaces of \(V\), let \(dim \;V = n\), and assume that \(dim \;U + dim \;W = n\).

1. If \(U \cap W = \{\mathbf{0}\}\), show that \(V = U \oplus W\).
2. If \(U + W = V\), show that \(V = U \oplus W\). [Hint: Theorem 6.4.5.]

Answer

2. By (a) it remains to show \(U + W = V\); we show that \(dim \;(U + W) = n\) and invoke Theorem 6.4.2. But \(U + W = U \oplus W\) because \(U \cap W = 0\), so \(dim \;(U + W) = dim \;U + dim \;W = n\).

Exercise \(\PageIndex{18}\)

Let \(A = \left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right]\) and consider \(T_{A} : \mathbb{R}^2 \to \mathbb{R}^2\).

1. Show that the only eigenvalue of \(T_{A}\) is \(\lambda = 0\).
2. Show that \(\text{ker }(T_A) = \mathbb{R}\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]\) is the unique \(T_{A}\)-invariant subspace of \(\mathbb{R}^2\) (except for \(0\) and \(\mathbb{R}^2\)).

Answer

2. First, \(\text{ker}(T_{A})\) is \(T_{A}\)-invariant. Let \(U = \mathbb{R}\mathbf{p}\) be \(T_{A}\)-invariant. Then \(T_{A}(\mathbf{p})\) is in \(U\), say \(T_{A}(\mathbf{p}) = \lambda\mathbf{p}\). Hence \(A\mathbf{p} = \lambda\mathbf{p}\) so \(\lambda\) is an eigenvalue of \(A\). This means that \(\lambda = 0\) by (a), so \(\mathbf{p}\) is in \(\text{ker}(T_{A})\). Thus \(U \subseteq \text{ker}(T_{A})\). But \(dim \;[\text{ker}(T_{A})] \neq 2\) because \(T_{A} \neq 0\), so \(dim \;[\text{ker}(T_{A})] = 1 = dim \;(U)\). Hence \(U = \text{ker}(T_{A})\).

Exercise \(\PageIndex{19}\)

If \(A = \left[ \begin{array}{rrrr} 2 & -5 & 0 & 0 \\ 1 & -2 & 0 & 0 \\ 0 & 0 & -1 & -2 \\ 0 & 0 & 1 & 1 \end{array} \right]\), show that \(T_{A} : \mathbb{R}^4 \to \mathbb{R}^4\) has two-dimensional \(T\)-invariant subspaces \(U\) and \(W\) such that \(\mathbb{R}^4 = U \oplus W\), but \(A\) has no real eigenvalue.

Exercise \(\PageIndex{20}\)

Let \(T : V \to V\) be a linear operator where \(dim \;V = n\). If \(U\) is a \(T\)-invariant subspace of \(V\), let \(T_{1} : U \to U\) denote the restriction of \(T\) to \(U\) (so \(T_{1}(\mathbf{u}) = T(\mathbf{u})\) for all \(\mathbf{u}\) in \(U\)). Show that \(c_T(x) = c_{T_1}(x) \cdot q(x)\) for some polynomial \(q(x)\). [Hint: Theorem 9.3.1.]

Answer

Let \(B_{1}\) be a basis of \(U\) and extend it to a basis \(B\) of \(V\). Then \(M_B(T) = \left[ \begin{array}{cc} M_{B_1}(T) & Y \\ 0 & Z \end{array} \right]\), so \(c_T(x) = \det [xI - M_B(T)] = \det [xI - M_{B_1}(T)]\det [xI - Z] = c_{T1}(x)q(x)\).

Exercise \(\PageIndex{21}\)

Let \(T : V \to V\) be a linear operator where \(dim \;V = n\). Show that \(V\) has a basis of eigenvectors if and only if \(V\) has a basis \(B\) such that \(M_{B}(T)\) is diagonal.

Exercise \(\PageIndex{22}\)

In each case, show that \(T^{2} = 1\) and find (as in Example 9.3.10) an ordered basis \(B\) such that \(M_{B}(T)\) has the given block form.

1. \(T :\|{M}_{22} \to\|{M}_{22}\) where \(T(A) = A^T\), \(M_B(T) = \left[ \begin{array}{rr} I_3 & 0 \\ 0 & -1 \end{array} \right]\)
2. \(T :\|{P}_3 \to\|{P}_3\) where \(T[p(x)] = p(-x)\), \(M_B(T) = \left[ \begin{array}{rr} I_2 & 0 \\ 0 & -I_2 \end{array} \right]\)
3. \(T : \mathbb{C} \to \mathbb{C}\) where \(T(a + bi) = a - bi\), \(M_B(T) = \left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right]\)
4. \(T : \mathbb{R}^3 \to \mathbb{R}^3\) where \(T(a, b, c) = (-a + 2b + c, b + c, -c)\), \(M_B(T) = \left[ \begin{array}{rr} 1 & 0 \\ 0 & -I_2 \end{array} \right]\)
5. \(T : V \to V\) where \(T(\mathbf{v}) = -\mathbf{v}\), \(dim \;V = n\), \(M_{B}(T) = -I_{n}\)

Answer

2. \(T^{2}[p(x)] = p[-(-x)] = p(x)\), so \(T^{2} = 1\);
4. \(B = \{1, x^{2};\ x, x^{3}\}\) \(T^{2}(a, b, c) = T(-a + 2b + c, b + c, -c) = (a, b, c)\), so \(T^{2} = 1\); \(B = \{(1, 1, 0);\ (1, 0, 0), (0, -1, 2)\}\)

Exercise \(\PageIndex{23}\)

Let \(U\) and \(W\) denote subspaces of a vector space \(V\).

1. If \(V = U \oplus W\), define \(T : V \to V\) by \(T(\mathbf{v}) = \mathbf{w}\) where \(\mathbf{v}\) is written (uniquely) as \(\mathbf{v} = \mathbf{u} + \mathbf{w}\) with \(\mathbf{u}\) in \(U\) and \(\mathbf{w}\) in \(W\). Show that \(T\) is a linear transformation, \(U = \text{ker }T\), \(W = im \;T\), and \(T^{2} = T\).
2. Conversely, if \(T : V \to V\) is a linear transformation such that \(T^{2} = T\), show that \(V = \text{ker }T \oplus im \;T\). [Hint: \(\mathbf{v} - T(\mathbf{v})\) lies in \(\text{ker }T\) for all \(\mathbf{v}\) in \(V\).] Use the Hint and Exercise 9.3.2.

Exercise \(\PageIndex{24}\)

Let \(T : V \to V\) be a linear operator satisfying \(T^{2} = T\) (such operators are called idempotents). Define \(U_{1} = \{\mathbf{v} \mid T(\mathbf{v}) = \mathbf{v}\}\), \(U_{2} = \text{ker }T = \{\mathbf{v} \mid T(\mathbf{v}) = 0\}\).

1. Show that \(V = U_{1} \oplus U_{2}\).
2. If \(dim \;V = n\), find a basis \(B\) of \(V\) such that \(M_B(T) = \left[ \begin{array}{rr} I_r & 0 \\ 0 & 0 \end{array} \right]\), where \(r = rank \;T\).
3. If \(A\) is an \(n \times n\) matrix such that \(A^{2} = A\), show that \(A\) is similar to \(\left[ \begin{array}{rr} I_r & 0 \\ 0 & 0 \end{array} \right]\), where \(r = rank \;A\).

Exercise \(\PageIndex{25}\)

In each case, show that \(T^{2} = T\) and find (as in the preceding exercise) an ordered basis \(B\) such that \(M_{B}(T)\) has the form given (\(0_{k}\) is the \(k \times k\) zero matrix).

1. \(T :\|{P}_2 \to\|{P}_2\) where \(T(a + bx + cx^2) = (a - b + c)(1 + x + x^2)\), \(M_B(T) = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0_2 \end{array} \right]\)
2. \(T : \mathbb{R}^3 \to \mathbb{R}^3\) where \(T(a, b, c) = (a + 2b, 0, 4b + c)\), \(M_B(T) = \left[ \begin{array}{rr} I_2 & 0 \\ 0 & 0 \end{array} \right]\)
3. \(T :\|{M}_{22} \to\|{M}_{22}\) where \(T\left[ \begin{array}{rr} a & b \\ c & d \end{array} \right] = \left[ \begin{array}{rr} -5 & -15 \\ 2 & 6 \end{array} \right]\left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]\), \(M_B(T) = \left[ \begin{array}{cc} I_2 & 0 \\ 0 & 0_2 \end{array} \right]\)

Answer

2. \(T^{2}(a, b, c) = T(a + 2b, 0, 4b + c) = (a + 2b, 0, 4b + c) = T(a, b, c)\), so \(T^{2} = T\); \(B = \{(1, 0, 0), (0, 0, 1);\ (2, -1, 4)\}\)

Exercise \(\PageIndex{26}\)

Let \(T : V \to V\) be an operator satisfying \(T^{2} = cT\), \(c \neq 0\).

1. Show that \(V = U \oplus \text{ker }T\), where \(U = \{\mathbf{u} \mid T(\mathbf{u}) = c\mathbf{u}\}\).
2. If \(dim \;V = n\), show that \(V\) has a basis \(B\) such that \(M_B(T) = \left[ \begin{array}{rr} cI_r & 0 \\ 0 & 0 \end{array} \right]\), where \(r = rank \;T\).
3. If \(A\) is any \(n \times n\) matrix of \(rank \;r\) such that \(A^{2} = cA\), \(c \neq 0\), show that \(A\) is similar to \(\left[ \begin{array}{rr} cI_r & 0 \\ 0 & 0 \end{array} \right]\).

Exercise \(\PageIndex{27}\)

Let \(T : V \to V\) be an operator such that \(T^{2} = c^{2}\), \(c \neq 0\).

1. Show that \(V = U_{1} \oplus U_{2}\), where \(U_{1} = \{\mathbf{v} \mid T(\mathbf{v}) = c\mathbf{v}\}\) and \(U_{2} = \{\mathbf{v} \mid T(\mathbf{v}) = -c\mathbf{v}\}\). [Hint: \(\mathbf{v} = \frac{1}{2c} \{[T(\mathbf{v}) + c\mathbf{v}] - [T(\mathbf{v}) - c\mathbf{v}]\}\).]
2. If \(dim \;V = n\), show that \(V\) has a basis \(B\) such that \(M_B(T) = \left[ \begin{array}{cc} cI_k & 0 \\ 0 & -cI_{n-k} \end{array} \right]\) for some \(k\).
3. If \(A\) is an \(n \times n\) matrix such that \(A^{2} = c^{2}I\), \(c \neq 0\), show that \(A\) is similar to \(\left[ \begin{array}{cc} cI_k & 0 \\ 0 & -cI_{n-k} \end{array} \right]\) for some \(k\).

Exercise \(\PageIndex{28}\)

If \(P\) is a fixed \(n \times n\) matrix, define \(T :\|{M}_{nn} \to\|{M}_{nn}\) by \(T(A) = PA\). Let \(U_{j}\) denote the subspace of \(\mathbf{M}_{nn}\) consisting of all matrices with all columns zero except possibly column \(j\).

1. Show that each \(U_{j}\) is \(T\)-invariant.
2. Show that \(\mathbf{M}_{nn}\) has a basis \(B\) such that \(M_{B}(T)\) is block diagonal with each block on the diagonal equal to \(P\).

Exercise \(\PageIndex{29}\)

Let \(V\) be a vector space. If \(f : V \to \mathbb{R}\) is a linear transformation and \(\mathbf{z}\) is a vector in \(V\), define \(T_{f,\mathbf{z}} : V \to V\) by \(T_{f,\mathbf{z}}(\mathbf{v}) = f(\mathbf{v})\mathbf{z}\) for all \(\mathbf{v}\) in \(V\). Assume that \(f \neq 0\) and \(\mathbf{z} \neq \mathbf{0}\).

1. Show that \(T_{f,\mathbf{z}}\) is a linear operator of \(rank \;1\).
2. If \(f \neq 0\), show that \(T_{f,\mathbf{z}}\) is an idempotent if and only if \(f(\mathbf{z}) = 1\). (Recall that \(T : V \to V\) is called an idempotent if \(T^{2} = T\).)
3. Show that every idempotent \(T : V \to V\) of \(rank \;1\) has the form \(T = T_{f,\mathbf{z}}\) for some \(f : V \to \mathbb{R}\) and some \(\mathbf{z}\) in \(V\) with \(f(\mathbf{z}) = 1\). [Hint: Write \(im \;T = \mathbb{R}\mathbf{z}\) and show that \(T(\mathbf{z}) = \mathbf{z}\). Then use Exercise 9.3.23.]

Answer

2. \(T_{f,\mathbf{z}}[T_{f,\mathbf{z}}(\mathbf{v})] = T_{f,\mathbf{z}}[f(\mathbf{v})\mathbf{z}] = f[f(\mathbf{v})\mathbf{z}]\mathbf{z} = f(\mathbf{v})\{f[\mathbf{z}]\mathbf{z}\} = f(\mathbf{v})f(\mathbf{z})\mathbf{z}\). This equals \(T_{f,\mathbf{z}}(\mathbf{v}) = f(\mathbf{v})\mathbf{z}\) for all \(\mathbf{v}\) if and only if \(f(\mathbf{v})f(\mathbf{z}) = f(\mathbf{v})\) for all \(\mathbf{v}\). Since \(f \neq 0\), this holds if and only if \(f(z) = 1\).

Exercise \(\PageIndex{30}\)

Let \(U\) be a fixed \(n \times n\) matrix, and consider the operator \(T :\|{M}_{nn} \to\|{M}_{nn}\) given by \(T(A) = UA\).

1. Show that \(\lambda\) is an eigenvalue of \(T\) if and only if it is an eigenvalue of \(U\).
2. If \(\lambda\) is an eigenvalue of \(T\), show that \(E_{\lambda}(T)\) consists of all matrices whose columns lie in \(E_{\lambda}(U)\): \(E_{\lambda}(T) \\ = \{ \left[ \begin{array}{cccc} P_1 & P_2 & \cdots & P_n \end{array} \right] \mid P_i \mbox{ in } E_{\lambda}(U) \mbox{ for each } i \}\)
3. Show if \(dim \;[E_{\lambda}(U)] = d\), then \(dim \;[E_{\lambda}(T)] = nd\). [Hint: If \(B = \{\mathbf{x}_{1}, \dots, \mathbf{x}_{d}\}\) is a basis of \(E_{\lambda}(U)\), consider the set of all matrices with one column from \(B\) and the other columns zero.]

Answer

2. If \(A = \left[ \begin{array}{cccc} \mathbf{p}_{1} & \mathbf{p}_2 & \cdots & \mathbf{p}_{n} \end{array} \right]\) where \(U\mathbf{p}_{i} = \lambda\mathbf{p}_{i}\) for each \(i\), then \(UA = \lambda A\). Conversely, \(UA = \lambda A\) means that \(U\mathbf{p} = \lambda\mathbf{p}\) for every column \(\mathbf{p}\) of \(A\).

Exercise \(\PageIndex{31}\)

Let \(T : V \to V\) be a linear operator where \(V\) is finite dimensional. If \(U \subseteq V\) is a subspace, let \(\overline{U} = \{ \mathbf{u}_0 + T(\mathbf{u}_1) + T^2(\mathbf{u}_2) + \cdots + T^k(\mathbf{u}_k) \mid \mathbf{u}_i \mbox{ in } U, k \geq 0 \}\). Show that \(\overline{U}\) is the smallest \(T\)-invariant subspace containing \(U\) (that is, it is \(T\)-invariant, contains \(U\), and is contained in every such subspace)

Exercise \(\PageIndex{32}\)

Let \(U_{1}, \dots, U_{m}\) be subspaces of \(V\) and assume that \(V = U_{1} + \cdots + U_{m}\); that is, every \(\mathbf{v}\) in \(V\) can be written (in at least one way) in the form \(\mathbf{v} = \mathbf{u}_{1} + \cdots + \mathbf{u}_{m}\), \(\mathbf{u}_{i}\) in \(U_{i}\). Show that the following conditions are equivalent.

1. If \(\mathbf{u}_{1} + \cdots + \mathbf{u}_{m} = \mathbf{0}\), \(\mathbf{u}_{i}\) in \(U_{i}\), then \(\mathbf{u}_{i} = \mathbf{0}\) for each \(i\).
2. If \(\mathbf{u}_{1} + \cdots + \mathbf{u}_{m} = \mathbf{u}^{\prime}_{1} + \cdots + \mathbf{u}^{\prime}_{m}\), \(\mathbf{u}_{i}\) and \(\mathbf{u}^{\prime}_{i}\) in \(U_{i}\), then \(\mathbf{u}_{i} = \mathbf{u}^{\prime}_{i}\) for each \(i\).
3. \(U_{i} \cap (U_{1} + \cdots + U_{i-1} + U_{i+1} + \cdots + U_{m}) = \{\mathbf{0}\}\) for each \(i = 1, 2, \dots, m\).
4. \(U_{i} \cap (U_{i+1} + \cdots + U_{m}) = \{\mathbf{0}\}\) for each \(i = 1, 2, \dots, m - 1\).

When these conditions are satisfied, we say that \(V\) is the direct sum of the subspaces \(U_{i}\), and write \(V = U_{1} \oplus U_{2} \oplus \cdots \oplus U_{m}\).

Exercise \(\PageIndex{33}\)

1. Let \(B\) be a basis of \(V\) and let \(B = B_{1} \cup B_{2} \cup \cdots \cup B_{m}\) where the \(B_{i}\) are pairwise disjoint, nonempty subsets of \(B\). If \(U_{i} = span \;B_{i}\) for each \(i\), show that \(V = U_{1} \oplus U_{2} \oplus \cdots \oplus U_{m}\) (preceding exercise).
2. Conversely if \(V = U_{1} \oplus \cdots \oplus U_{m}\) and \(B_{i}\) is a basis of \(U_{i}\) for each \(i\), show that \(B = B_{1} \cup \cdots \cup B_{m}\) is a basis of \(V\) as in (a).

Answer

2. Let \(T: V \to V\) be an operator where \(T^3=0\). If \(\mathbf{u} \in V\) and \(U = span \;\{\mathbf{u},T(\mathbf{u}), T^2(\mathbf{u})\}\), show that \(U\) is \(T\)-invariant and has dimension \(3\).