3-6. Consider IR3 with the inmer product
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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![### Topic: Orthogonal Projections and Inner Products in \(\mathbb{R}^3\)
#### Problem Statement
Consider \(\mathbb{R}^3\) with the inner product defined as:
\[
\langle x, y \rangle := x^T
\begin{bmatrix}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
y
\]
Furthermore, we define \(e_1, e_2, e_3\) as the standard (canonical) basis in \(\mathbb{R}^3\).
#### Tasks
**a.** Determine the orthogonal projection \(\pi_U(e_2)\) of \(e_2\) onto \(U = \text{span}\{e_1, e_3\}\).
> *Hint:* Orthogonality is defined through the inner product.
**b.** Compute the distance \(d(e_2, U)\).
**c.** Draw the scenario: standard basis vectors and \(\pi_U(e_2)\).
#### Explanation
- The **inner product** given here is defined by the matrix:
\[
\begin{bmatrix}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]
This modifies the standard dot product in \(\mathbb{R}^3\), which can affect geometric interpretations like lengths and angles.
- **Orthogonal Projection**: The task involves projecting \(e_2\) onto a subspace \(U\) spanned by two basis vectors \(e_1\) and \(e_3\). The orthogonal projection minimizes the distance from the vector to the subspace.
- **Distance Calculation**: Needed after finding the projection, this represents how far \(e_2\) is from \(U\).
- **Diagram Explanation**: Drawing this scenario involves illustrating the basis vectors \(e_1, e_2, e_3\) and showing where \(e_2\) projects onto \(U\) along with the calculated distance.
This exercise enhances understanding of projection operations in vector spaces using different inner products.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F2c3d016e-d608-4eca-8ae8-002575c35941%2Fcbp7bni_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Orthogonal Projections and Inner Products in \(\mathbb{R}^3\)
#### Problem Statement
Consider \(\mathbb{R}^3\) with the inner product defined as:
\[
\langle x, y \rangle := x^T
\begin{bmatrix}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
y
\]
Furthermore, we define \(e_1, e_2, e_3\) as the standard (canonical) basis in \(\mathbb{R}^3\).
#### Tasks
**a.** Determine the orthogonal projection \(\pi_U(e_2)\) of \(e_2\) onto \(U = \text{span}\{e_1, e_3\}\).
> *Hint:* Orthogonality is defined through the inner product.
**b.** Compute the distance \(d(e_2, U)\).
**c.** Draw the scenario: standard basis vectors and \(\pi_U(e_2)\).
#### Explanation
- The **inner product** given here is defined by the matrix:
\[
\begin{bmatrix}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]
This modifies the standard dot product in \(\mathbb{R}^3\), which can affect geometric interpretations like lengths and angles.
- **Orthogonal Projection**: The task involves projecting \(e_2\) onto a subspace \(U\) spanned by two basis vectors \(e_1\) and \(e_3\). The orthogonal projection minimizes the distance from the vector to the subspace.
- **Distance Calculation**: Needed after finding the projection, this represents how far \(e_2\) is from \(U\).
- **Diagram Explanation**: Drawing this scenario involves illustrating the basis vectors \(e_1, e_2, e_3\) and showing where \(e_2\) projects onto \(U\) along with the calculated distance.
This exercise enhances understanding of projection operations in vector spaces using different inner products.
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