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2) Conversely Suppose that V = UW and that T is the projection on a U along W. Prove that there is a basis such that the matrix of T is a diagonal matrix with only 0 and 1 on the diagonal.

2) Conversely Suppose that V = UW and that T is the projection on a U along W. Prove that there is a basis such that the matrix of T is a diagonal matrix with only 0 and 1 on the diagonal.

Advanced Engineering Mathematics
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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### Linear Algebra Problem: Diagonalization of a Projection Matrix 2) *Conversely:* Suppose that \( V = U \oplus W \) and that \( T \) is the projection on \( U \) along \( W \). Prove that there is a basis such that the matrix of \( T \) is a diagonal matrix with only 0 and 1 on the diagonal. #### Solution Outline: 1. **Understand the given projection:** Given \( V \) can be decomposed into the direct sum of two subspaces \( U \) and \( W \), i.e., \( V = U \oplus W \). Here, \( \oplus \) denotes the direct sum. 2. **Projection Definition:** The linear transformation \( T \) is the projection operator onto \( U \), meaning \( T: V \rightarrow U \). Any vector in \( V \) can be written as \( v = u + w \) where \( u \in U \) and \( w \in W \). 3. **Action of T:** When \( T \) is applied to \( v = u + w \), it should return the vector \( u \), i.e., \( T(v) = u \). 4. **Basis Selection:** Construct a basis for \( V \) by taking the union of bases for \( U \) and \( W \). Let \( \{u_1, u_2, ..., u_m\} \) be a basis for \( U \) and \( \{w_1, w_2, ..., w_n\} \) be a basis for \( W \). This set \( B = \{u_1, u_2, ..., u_m, w_1, w_2, ..., w_n\} \) forms a basis for \( V \). 5. **Matrix Representation of T:** In the basis \( B \), the projection matrix \( T \) maps each basis vector to itself if it's in \( U \) and to zero if it's in \( W \). Thus, the matrix of \( T \) in this basis is a diagonal matrix with 1's corresponding to basis vectors in \( U \) and 0's corresponding to basis vectors in \( W \). 6. **Diagonal Matrix:** Explicitly, the matrix representation
Transcribed Image Text:### Linear Algebra Problem: Diagonalization of a Projection Matrix 2) *Conversely:* Suppose that \( V = U \oplus W \) and that \( T \) is the projection on \( U \) along \( W \). Prove that there is a basis such that the matrix of \( T \) is a diagonal matrix with only 0 and 1 on the diagonal. #### Solution Outline: 1. **Understand the given projection:** Given \( V \) can be decomposed into the direct sum of two subspaces \( U \) and \( W \), i.e., \( V = U \oplus W \). Here, \( \oplus \) denotes the direct sum. 2. **Projection Definition:** The linear transformation \( T \) is the projection operator onto \( U \), meaning \( T: V \rightarrow U \). Any vector in \( V \) can be written as \( v = u + w \) where \( u \in U \) and \( w \in W \). 3. **Action of T:** When \( T \) is applied to \( v = u + w \), it should return the vector \( u \), i.e., \( T(v) = u \). 4. **Basis Selection:** Construct a basis for \( V \) by taking the union of bases for \( U \) and \( W \). Let \( \{u_1, u_2, ..., u_m\} \) be a basis for \( U \) and \( \{w_1, w_2, ..., w_n\} \) be a basis for \( W \). This set \( B = \{u_1, u_2, ..., u_m, w_1, w_2, ..., w_n\} \) forms a basis for \( V \). 5. **Matrix Representation of T:** In the basis \( B \), the projection matrix \( T \) maps each basis vector to itself if it's in \( U \) and to zero if it's in \( W \). Thus, the matrix of \( T \) in this basis is a diagonal matrix with 1's corresponding to basis vectors in \( U \) and 0's corresponding to basis vectors in \( W \). 6. **Diagonal Matrix:** Explicitly, the matrix representation
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