Question
upstudy study bank question image url

Determine whether the functions \( \mathrm{y}_{1} \) and \( \mathrm{y}_{2} \) are linearly dependent on the interval \( (0,1) \). \[ y_{1}=1-2 \sin ^{2} t \cdot y_{2}=6 \cos 2 t \] Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Since \( y_{1}= \) \( \square \) \( y_{2} \) on \( (0,1) \), the functions are linearly independent on \( (0,1) \). (Simplify your answer.) B. Since \( y_{1}=\left(\square y_{2}\right. \) on \( (0,1) \), the functions are linearly dependent on \( (0,1) \). (Simplify your answer.) \( \square \) (Simplify your answer.) C. Since \( y_{1} \) is not a constant multiple of \( y_{2} \) on ( 0,1 ). the functions are linearly independent on ( 0,1 ). D. Since \( y_{1} \) is not a constant multiple of \( y_{2} \) on ( 0,1 ). the functions are linearly dependent on \( (0,1) \).

Ask by Matthews Greene. in the United States
Mar 17,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The functions \( y_1 \) and \( y_2 \) are linearly dependent on the interval \( (0,1) \) because \( y_1 = \frac{1}{6} y_2 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To determine whether the functions \( y_{1} = 1 - 2 \sin^2 t \) and \( y_{2} = 6 \cos 2t \) are linearly dependent or independent, we can express \( y_1 \) in terms of \( y_2 \). We know that \( \cos 2t = 1 - 2 \sin^2 t \), so we can rewrite \( y_{1} \) as \( y_{1} = 1 - 2 \sin^2 t = \cos 2t \). Thus, \( y_{1} = \frac{1}{6} y_{2} \) on \( (0,1) \), making them linearly dependent. So, we can fill in the blanks as follows: B. Since \( y_{1}=\left(\frac{1}{6} y_{2}\right) \) on \( (0,1) \), the functions are linearly dependent on \( (0,1) \). \( \frac{1}{6} \)

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy