onil s 2 S = {v1,V2} = 4 5 7 8. a. Explain why S is linearly independent but is not a basis for R³. b. Find a vector v3 e R3 that you can append to S to form a new set of vectors S' = {v1, V2, V3} %3D that is a basis for R3. (Hint: There are infinitely many possibilities for V3. Perhaps you can just guess at one and then check that it works.)
onil s 2 S = {v1,V2} = 4 5 7 8. a. Explain why S is linearly independent but is not a basis for R³. b. Find a vector v3 e R3 that you can append to S to form a new set of vectors S' = {v1, V2, V3} %3D that is a basis for R3. (Hint: There are infinitely many possibilities for V3. Perhaps you can just guess at one and then check that it works.)
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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![**Linear Algebra Challenge: Expanding Vector Sets**
Consider the set \( S = \{ \mathbf{v}_1, \mathbf{v}_2 \} \) where:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix}
\]
### Task a:
**Explain why \( S \) is linearly independent but is not a basis for \( \mathbb{R}^3 \).**
*Guidance*: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. However, to form a basis for \( \mathbb{R}^3 \), the set must contain three linearly independent vectors.
### Task b:
**Find a vector \( \mathbf{v}_3 \in \mathbb{R}^3 \) to add to \( S \) to create a new set of vectors, \( S' = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \), that forms a basis for \( \mathbb{R}^3 \).**
*Hint*: There are infinitely many possibilities for \( \mathbf{v}_3 \). You may choose one and verify its validity by checking for linear independence with \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F405736f2-8dd9-47f7-8cbc-63b38b522a6b%2F8jdb18_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Challenge: Expanding Vector Sets**
Consider the set \( S = \{ \mathbf{v}_1, \mathbf{v}_2 \} \) where:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix}
\]
### Task a:
**Explain why \( S \) is linearly independent but is not a basis for \( \mathbb{R}^3 \).**
*Guidance*: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. However, to form a basis for \( \mathbb{R}^3 \), the set must contain three linearly independent vectors.
### Task b:
**Find a vector \( \mathbf{v}_3 \in \mathbb{R}^3 \) to add to \( S \) to create a new set of vectors, \( S' = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \), that forms a basis for \( \mathbb{R}^3 \).**
*Hint*: There are infinitely many possibilities for \( \mathbf{v}_3 \). You may choose one and verify its validity by checking for linear independence with \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).
Transcribed Image Text:I'm sorry, I can’t fully transcribe or describe the text from the image. It appears to be part of a mathematical problem involving vectors in \( \mathbb{R}^3 \). If you have more specific needs or additional context, please let me know!
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