Find the B-coordinate vector of x E H where B = {b₁,b₂} is a basis of subspace H. b₁ = H b₂ = [6] 3 2 x = 2 图
Find the B-coordinate vector of x E H where B = {b₁,b₂} is a basis of subspace H. b₁ = H b₂ = [6] 3 2 x = 2 图
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
![**Problem Statement:**
Find the \(\mathcal{B}\)-coordinate vector of \( x \in H \) where \(\mathcal{B} = \{b_1, b_2\}\) is a basis of subspace \( H \).
**Given Vectors:**
\[ b_1 = \begin{bmatrix} 1 \\ -5 \\ -4 \end{bmatrix}, \quad b_2 = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \quad x = \begin{bmatrix} 5 \\ 2 \\ 1 \end{bmatrix} \]
**Explanation:**
The task is to express vector \( x \) as a linear combination of the basis vectors \( b_1 \) and \( b_2 \). This involves finding scalars \( c_1 \) and \( c_2 \) such that:
\[ x = c_1 b_1 + c_2 b_2 \]
Solving this equation will give the \(\mathcal{B}\)-coordinate vector for \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30f79184-3047-455a-b3d8-f2ad47623cdf%2Fb17f6277-e30f-422d-9559-dbb9b6ebc466%2Ftq7dppf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the \(\mathcal{B}\)-coordinate vector of \( x \in H \) where \(\mathcal{B} = \{b_1, b_2\}\) is a basis of subspace \( H \).
**Given Vectors:**
\[ b_1 = \begin{bmatrix} 1 \\ -5 \\ -4 \end{bmatrix}, \quad b_2 = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \quad x = \begin{bmatrix} 5 \\ 2 \\ 1 \end{bmatrix} \]
**Explanation:**
The task is to express vector \( x \) as a linear combination of the basis vectors \( b_1 \) and \( b_2 \). This involves finding scalars \( c_1 \) and \( c_2 \) such that:
\[ x = c_1 b_1 + c_2 b_2 \]
Solving this equation will give the \(\mathcal{B}\)-coordinate vector for \( x \).
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