Question - s be the subspace ob R spanned by the (x,M»,' and y=Cy, Yo, y,)' Let A = show that = N CA).
Question - s be the subspace ob R spanned by the (x,M»,' and y=Cy, Yo, y,)' Let A = show that = N CA).
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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Question
![**Question:**
Let \( S \) be the subspace of \( \mathbb{R}^3 \) spanned by the vectors \( x = (x_1, x_2, x_3)^T \) and \( y = (y_1, y_2, y_3)^T \).
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3
\end{bmatrix}
\]
Show that \( S^\perp = \mathcal{N}(\text{CA}) \).
**Explanation:**
- \( S \) is a subspace in \(\mathbb{R}^3\) formed by two vectors.
- \( S^\perp \) denotes the orthogonal complement of \( S \).
- \(\mathcal{N}(\text{CA})\) represents the null space of the given matrix \( A \).
- This involves linear algebra concepts such as vector spaces, orthogonality, and null spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc40ed8c4-e2aa-4488-bef1-34c2accebf46%2F68b1cde3-ecae-4168-b781-6959ccfa8bb1%2Fwe649w_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Let \( S \) be the subspace of \( \mathbb{R}^3 \) spanned by the vectors \( x = (x_1, x_2, x_3)^T \) and \( y = (y_1, y_2, y_3)^T \).
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3
\end{bmatrix}
\]
Show that \( S^\perp = \mathcal{N}(\text{CA}) \).
**Explanation:**
- \( S \) is a subspace in \(\mathbb{R}^3\) formed by two vectors.
- \( S^\perp \) denotes the orthogonal complement of \( S \).
- \(\mathcal{N}(\text{CA})\) represents the null space of the given matrix \( A \).
- This involves linear algebra concepts such as vector spaces, orthogonality, and null spaces.
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