Show that the vectors 3 Ot span R³ by giving a vector not in their span. B00 4
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
![**Exercise:**
Show that the vectors
\[
\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 4 \\ 1 \end{bmatrix}
\]
do not span \(\mathbb{R}^3\) by giving a vector not in their span.
\[
\begin{bmatrix} \ \\ \ \\ \ \end{bmatrix}
\]
**Explanation:**
In this exercise, you are asked to demonstrate that the set of vectors provided does not have the capability to span the entire three-dimensional space, \(\mathbb{R}^3\). A set of vectors spans \(\mathbb{R}^3\) if any vector in \(\mathbb{R}^3\) can be expressed as a linear combination of these vectors. By identifying a vector that cannot be expressed in this way, you prove that the span is insufficient. Fill in the matrix with an example of such a vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2F6afdbcc2-47a2-43d3-82ab-3175be05abe7%2Fo55uvf_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise:**
Show that the vectors
\[
\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 4 \\ 1 \end{bmatrix}
\]
do not span \(\mathbb{R}^3\) by giving a vector not in their span.
\[
\begin{bmatrix} \ \\ \ \\ \ \end{bmatrix}
\]
**Explanation:**
In this exercise, you are asked to demonstrate that the set of vectors provided does not have the capability to span the entire three-dimensional space, \(\mathbb{R}^3\). A set of vectors spans \(\mathbb{R}^3\) if any vector in \(\mathbb{R}^3\) can be expressed as a linear combination of these vectors. By identifying a vector that cannot be expressed in this way, you prove that the span is insufficient. Fill in the matrix with an example of such a vector.
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