Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Vectors in Linear Algebra
Consider the vectors **v₁**, **v₂**, **v₃**, and **b** given below:
\[
v₁ = \begin{bmatrix}
1 \\
4 \\
6
\end{bmatrix}, \quad v₂ = \begin{bmatrix}
2 \\
5 \\
7
\end{bmatrix}, \quad v₃ = \begin{bmatrix}
3 \\
6 \\
8
\end{bmatrix}, \quad b = \begin{bmatrix}
4 \\
7 \\
9
\end{bmatrix}
\]
In these vectors:
- **v₁** is a 3-dimensional vector with elements 1, 4, and 6.
- **v₂** is a 3-dimensional vector with elements 2, 5, and 7.
- **v₃** is a 3-dimensional vector with elements 3, 6, and 8.
- **b** is a 3-dimensional vector with elements 4, 7, and 9.
These vectors can be used in various linear algebra operations such as vector addition, scalar multiplication, dot product, and cross product, depending on the context and the dimensions involved. They can also be part of systems of linear equations, matrix operations, and transformations in higher-dimensional space.](https://content.bartleby.com/qna-images/question/ee22cf2f-b974-4b00-a3cf-09b388e7d65d/b4f9405c-82e6-4580-9a73-f0233e78f2ac/4lsa6g_processed.jpeg)
Transcribed Image Text:### Vectors in Linear Algebra
Consider the vectors **v₁**, **v₂**, **v₃**, and **b** given below:
\[
v₁ = \begin{bmatrix}
1 \\
4 \\
6
\end{bmatrix}, \quad v₂ = \begin{bmatrix}
2 \\
5 \\
7
\end{bmatrix}, \quad v₃ = \begin{bmatrix}
3 \\
6 \\
8
\end{bmatrix}, \quad b = \begin{bmatrix}
4 \\
7 \\
9
\end{bmatrix}
\]
In these vectors:
- **v₁** is a 3-dimensional vector with elements 1, 4, and 6.
- **v₂** is a 3-dimensional vector with elements 2, 5, and 7.
- **v₃** is a 3-dimensional vector with elements 3, 6, and 8.
- **b** is a 3-dimensional vector with elements 4, 7, and 9.
These vectors can be used in various linear algebra operations such as vector addition, scalar multiplication, dot product, and cross product, depending on the context and the dimensions involved. They can also be part of systems of linear equations, matrix operations, and transformations in higher-dimensional space.
![**Question:**
Is the set of vectors S = {v₁, v₂, v₃} linearly independent? Explain.
---
**Explanation:**
To determine whether the set of vectors \( S = \{v_1, v_2, v_3\} \) is linearly independent, we need to assess whether the only solution to the equation
\[ c_1 v_1 + c_2 v_2 + c_3 v_3 = 0 \]
is the trivial solution where \( c_1 = 0 \), \( c_2 = 0 \), and \( c_3 = 0 \).
1. Formulate the vectors in matrix form.
2. Perform row-reduction (Gaussian elimination) on the matrix formed by these vectors.
3. If, after row-reduction, the matrix has a pivot (leading 1) in every column, then the vectors are linearly independent.
4. If any column does not have a pivot, the vectors are linearly dependent.
A step-by-step matrix representation and row-reduction would clearly demonstrate the linear dependence or independence of the vectors. Ensure to provide the vectors' coordinates or their explicit representations for a detailed verification.](https://content.bartleby.com/qna-images/question/ee22cf2f-b974-4b00-a3cf-09b388e7d65d/b4f9405c-82e6-4580-9a73-f0233e78f2ac/si6ulln_processed.jpeg)
Transcribed Image Text:**Question:**
Is the set of vectors S = {v₁, v₂, v₃} linearly independent? Explain.
---
**Explanation:**
To determine whether the set of vectors \( S = \{v_1, v_2, v_3\} \) is linearly independent, we need to assess whether the only solution to the equation
\[ c_1 v_1 + c_2 v_2 + c_3 v_3 = 0 \]
is the trivial solution where \( c_1 = 0 \), \( c_2 = 0 \), and \( c_3 = 0 \).
1. Formulate the vectors in matrix form.
2. Perform row-reduction (Gaussian elimination) on the matrix formed by these vectors.
3. If, after row-reduction, the matrix has a pivot (leading 1) in every column, then the vectors are linearly independent.
4. If any column does not have a pivot, the vectors are linearly dependent.
A step-by-step matrix representation and row-reduction would clearly demonstrate the linear dependence or independence of the vectors. Ensure to provide the vectors' coordinates or their explicit representations for a detailed verification.
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