1. DETAILS POOLELINALG4 6.2.012. MYΝΟΤES ASK YOUR TEACHER Test the set of functions for linear independence in F. If it is linearly dependent, express one of the functions as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables fand g as they relate to the question.) () = e, g(x) =e 2. DETAILS POOLELINALG4 6.2.018. MY NOTES ASK YOUR TEACHER Determine whether the set 5 is a basis for the vector space V. O Bis a basis for v. O Bis not a basis for V.
1. DETAILS POOLELINALG4 6.2.012. MYΝΟΤES ASK YOUR TEACHER Test the set of functions for linear independence in F. If it is linearly dependent, express one of the functions as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables fand g as they relate to the question.) () = e, g(x) =e 2. DETAILS POOLELINALG4 6.2.018. MY NOTES ASK YOUR TEACHER Determine whether the set 5 is a basis for the vector space V. O Bis a basis for v. O Bis not a basis for V.
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 1: POOLLLINALG4 6.e.012**
Test the set of functions for linear independence in \( \mathbb{F} \). If it is linearly dependent, express one of the functions as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables \( f \) and \( g \) as they relate to the question.)
Functions:
- \( f(t) = e^t \)
- \( g(t) = e^{-t} \)
---
**Question 2: POOLLLINALG4 6.e.018**
Determine whether the set \( B \) is a basis for the vector space \( V \).
\[ V = M_{2 \times 2} \]
\[ B = \left\{ \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix} \right\} \]
Options:
- \( B \) is a basis for \( V \).
- \( B \) is not a basis for \( V \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d702b96-16c9-42c6-953e-90c469a3b56f%2F4e13e8a9-4752-412f-86cd-76b661e8e6ba%2Fk0nsy5d_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 1: POOLLLINALG4 6.e.012**
Test the set of functions for linear independence in \( \mathbb{F} \). If it is linearly dependent, express one of the functions as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables \( f \) and \( g \) as they relate to the question.)
Functions:
- \( f(t) = e^t \)
- \( g(t) = e^{-t} \)
---
**Question 2: POOLLLINALG4 6.e.018**
Determine whether the set \( B \) is a basis for the vector space \( V \).
\[ V = M_{2 \times 2} \]
\[ B = \left\{ \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix} \right\} \]
Options:
- \( B \) is a basis for \( V \).
- \( B \) is not a basis for \( V \).
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