Let v, V3= , and let H be the set of vectors in R° whose second and third entries are equal. Then every vector in H has a unique expansion as a linear combination of v1, V2, and v3 because the following equation is true for any s and t + (t-s) Is the set S= {v1, V2, V3) a basis for H? Why or why not? Choose the correct answer below. O A. The set S is not a basis for H because v, V2, and va are linearly dependent. O B. The set S is not a basis for H because there are elements in Span/v, v2, V3) that are not in H. OC. The set S is a basis for H because v,, V2, and v3 are linearly independent and span H. O D. The set S is a basis for H because every element in H is a linear combination of the vectors in S.
Let v, V3= , and let H be the set of vectors in R° whose second and third entries are equal. Then every vector in H has a unique expansion as a linear combination of v1, V2, and v3 because the following equation is true for any s and t + (t-s) Is the set S= {v1, V2, V3) a basis for H? Why or why not? Choose the correct answer below. O A. The set S is not a basis for H because v, V2, and va are linearly dependent. O B. The set S is not a basis for H because there are elements in Span/v, v2, V3) that are not in H. OC. The set S is a basis for H because v,, V2, and v3 are linearly independent and span H. O D. The set S is a basis for H because every element in H is a linear combination of the vectors in S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \), \( \mathbf{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \), and let \( H \) be the set of vectors in \( \mathbb{R}^3 \) whose second and third entries are equal. Then every vector in \( H \) has a unique expansion as a linear combination of \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \) because the following equation is true for any \( s \) and \( t \):
\[
s \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + t \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} s \\ t \\ s + t \end{bmatrix} = (t-s) \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + s \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}
\]
Is the set \( S = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) a basis for \( H \)? Why or why not?
Choose the correct answer below.
- \( \circ \) A. The set \( S \) is not a basis for \( H \) because \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \) are linearly dependent.
- \( \circ \) B. The set \( S \) is not a basis for \( H \) because there are elements in \( \text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) \) that are not in \( H \).
- \( \circ \) C. The set \( S \) is a basis for \( H \) because \( \mathbf{v}_1, \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6b254a-d25d-479c-b42b-752b8b5bbcbb%2Fb8b0bc98-a548-4908-89ad-66a160127b3f%2F7gzr1ag_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \), \( \mathbf{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \), and let \( H \) be the set of vectors in \( \mathbb{R}^3 \) whose second and third entries are equal. Then every vector in \( H \) has a unique expansion as a linear combination of \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \) because the following equation is true for any \( s \) and \( t \):
\[
s \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + t \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} s \\ t \\ s + t \end{bmatrix} = (t-s) \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + s \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}
\]
Is the set \( S = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) a basis for \( H \)? Why or why not?
Choose the correct answer below.
- \( \circ \) A. The set \( S \) is not a basis for \( H \) because \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \) are linearly dependent.
- \( \circ \) B. The set \( S \) is not a basis for \( H \) because there are elements in \( \text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) \) that are not in \( H \).
- \( \circ \) C. The set \( S \) is a basis for \( H \) because \( \mathbf{v}_1, \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
