Let T : R → R³ be the orthogonal projection onto the line L in R3 containing all scalar multiples of the vector 1 (a) Express T in matrix form using a 3x3 matrix A. (b) Find the rank of the matrix A.
Let T : R → R³ be the orthogonal projection onto the line L in R3 containing all scalar multiples of the vector 1 (a) Express T in matrix form using a 3x3 matrix A. (b) Find the rank of the matrix A.
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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![**Orthogonal Projection in \(\mathbb{R}^3\)**
Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be the orthogonal projection onto the line \(\mathcal{L}\) in \(\mathbb{R}^3\) containing all scalar multiples of the vector
\[
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}
\]
**Tasks:**
(a) Express \( T \) in matrix form using a \( 3 \times 3 \) matrix \( A \).
(b) Find the rank of the matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b1372ae-e668-4971-89ee-29da7ac7c466%2Ff2cd39b1-38d0-41ad-bf9a-3a645e839645%2Fiz2kza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Orthogonal Projection in \(\mathbb{R}^3\)**
Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be the orthogonal projection onto the line \(\mathcal{L}\) in \(\mathbb{R}^3\) containing all scalar multiples of the vector
\[
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}
\]
**Tasks:**
(a) Express \( T \) in matrix form using a \( 3 \times 3 \) matrix \( A \).
(b) Find the rank of the matrix \( A \).
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