1) Determine if the vector b can be written as a linear combination of the vectors a₁, az, and al 1 2. [] 1 93 허
1) Determine if the vector b can be written as a linear combination of the vectors a₁, az, and al 1 2. [] 1 93 허
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 1:** Determine if the vector **b** can be written as a linear combination of the vectors **a₁**, **a₂**, and **a₃**.
Vectors:
\[
\vec{a_1} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad
\vec{a_2} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad
\vec{a_3} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad
\vec{b} = \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}
\]
**Explanation:**
The question asks whether the vector **b** can be expressed as a linear combination of **a₁**, **a₂**, and **a₃**. This means we need to determine if there exist scalars **x₁**, **x₂**, and **x₃** such that:
\[
\vec{b} = x_1\vec{a_1} + x_2\vec{a_2} + x_3\vec{a_3}
\]
Substituting the vectors into the equation, we have:
\[
\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix} = x_1 \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}
\]
This can be expanded to the following system of equations:
1. \( x_1 = 2 \)
2. \( x_2 = -1 \)
3. \( x_3 = 2 \)
These equations clearly show that each component of **b** can be achieved by adjusting the coefficients of **a₁**, **a₂**, and **a₃**, indicating that **b** can indeed be expressed as a linear combination of these vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa589a0b1-5940-47b0-b798-8fa24b0ff23c%2F99f0076f-0255-4bcf-a051-84c3f0e37044%2F9llsafq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1:** Determine if the vector **b** can be written as a linear combination of the vectors **a₁**, **a₂**, and **a₃**.
Vectors:
\[
\vec{a_1} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad
\vec{a_2} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad
\vec{a_3} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad
\vec{b} = \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}
\]
**Explanation:**
The question asks whether the vector **b** can be expressed as a linear combination of **a₁**, **a₂**, and **a₃**. This means we need to determine if there exist scalars **x₁**, **x₂**, and **x₃** such that:
\[
\vec{b} = x_1\vec{a_1} + x_2\vec{a_2} + x_3\vec{a_3}
\]
Substituting the vectors into the equation, we have:
\[
\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix} = x_1 \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}
\]
This can be expanded to the following system of equations:
1. \( x_1 = 2 \)
2. \( x_2 = -1 \)
3. \( x_3 = 2 \)
These equations clearly show that each component of **b** can be achieved by adjusting the coefficients of **a₁**, **a₂**, and **a₃**, indicating that **b** can indeed be expressed as a linear combination of these vectors.
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