Suppose that V is a vector space over the scalars from R, and T : V → V is a linear transformation from V to V. Let U be a subspace of V. We say that U is T-invariant if it satisfies the following condition: For every u EU, T(u) is also in U. (a) For subspaces U₁ and U₂ of V, we define U₁+U₂ as follows: U₁+U₂ = {u₁ + U₂ | µ₁ € V₁, U₂ € U₂} Suppose that U₁ and U₂ are both T-invariant. Prove that U₁+U₂ is also T-invariant. (b) Now suppose that S and T are both linear transformations from V to V that satisfy the condition SOT = TOS where (SoT)(v) = S(T(v)) for every v V. Prove the following statement: Ker(T-XI) is S-invariant for every > € R.
Suppose that V is a vector space over the scalars from R, and T : V → V is a linear transformation from V to V. Let U be a subspace of V. We say that U is T-invariant if it satisfies the following condition: For every u EU, T(u) is also in U. (a) For subspaces U₁ and U₂ of V, we define U₁+U₂ as follows: U₁+U₂ = {u₁ + U₂ | µ₁ € V₁, U₂ € U₂} Suppose that U₁ and U₂ are both T-invariant. Prove that U₁+U₂ is also T-invariant. (b) Now suppose that S and T are both linear transformations from V to V that satisfy the condition SOT = TOS where (SoT)(v) = S(T(v)) for every v V. Prove the following statement: Ker(T-XI) is S-invariant for every > € R.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Suppose that \( V \) is a vector space over the scalars from \( \mathbb{R} \), and \( T: V \rightarrow V \) is a linear transformation from \( V \) to \( V \). Let \( U \) be a subspace of \( V \). We say that \( U \) is \( T \)-invariant if it satisfies the following condition:
For every \( u \in U \), \( T(u) \) is also in \( U \).
(a) For subspaces \( U_1 \) and \( U_2 \) of \( V \), we define \( U_1 + U_2 \) as follows:
\[ U_1 + U_2 := \{ \mathbf{u_1} + \mathbf{u_2} \, | \, \mathbf{u_1} \in U_1, \, \mathbf{u_2} \in U_2 \} \]
Suppose that \( U_1 \) and \( U_2 \) are both \( T \)-invariant. Prove that \( U_1 + U_2 \) is also \( T \)-invariant.
(b) Now suppose that \( S \) and \( T \) are both linear transformations from \( V \) to \( V \) that satisfy the condition
\[ S \circ T = T \circ S \]
where \( (S \circ T)(\mathbf{v}) = S(T(\mathbf{v})) \) for every \( \mathbf{v} \in V \). Prove the following statement:
\[ \text{Ker}(T - \lambda I) \text{ is } S \text{-invariant for every } \lambda \in \mathbb{R}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6e6d380-54a4-46c8-adc0-56f54d5d0909%2F01e16a22-d4ad-4773-90e3-eeac38352a83%2Fqys6yv9_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that \( V \) is a vector space over the scalars from \( \mathbb{R} \), and \( T: V \rightarrow V \) is a linear transformation from \( V \) to \( V \). Let \( U \) be a subspace of \( V \). We say that \( U \) is \( T \)-invariant if it satisfies the following condition:
For every \( u \in U \), \( T(u) \) is also in \( U \).
(a) For subspaces \( U_1 \) and \( U_2 \) of \( V \), we define \( U_1 + U_2 \) as follows:
\[ U_1 + U_2 := \{ \mathbf{u_1} + \mathbf{u_2} \, | \, \mathbf{u_1} \in U_1, \, \mathbf{u_2} \in U_2 \} \]
Suppose that \( U_1 \) and \( U_2 \) are both \( T \)-invariant. Prove that \( U_1 + U_2 \) is also \( T \)-invariant.
(b) Now suppose that \( S \) and \( T \) are both linear transformations from \( V \) to \( V \) that satisfy the condition
\[ S \circ T = T \circ S \]
where \( (S \circ T)(\mathbf{v}) = S(T(\mathbf{v})) \) for every \( \mathbf{v} \in V \). Prove the following statement:
\[ \text{Ker}(T - \lambda I) \text{ is } S \text{-invariant for every } \lambda \in \mathbb{R}. \]
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