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Consider the following statements about the linear dependence of the real valued functions y1 = 1, y2 = x and y3 = x2, over the field of real numbers.
I. y1, y2 and y3 are linearly independent on – 1 ≤ x ≤ 0
II. y1, y2 and y3 are linearly dependent on 0 ≤ x ≤ 1
III. y1, y2 and y3 are linearly independent on 0 ≤ x ≤ 1
IV. y1, y2 and y3 are linearly dependent on – 1 ≤ x ≤ 0
Which one among the following is correct?
- a)Both I and II are true
- b)Both I and III are true
- c)Both II and IV are true
- d)Both III and IV are true
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Consider the following statements about the linear dependence of the r...
Concept:
The linear dependency or Independency of function fi can be found using Wronskion Matrix.
An analytic function is linearly dependent if:
Application:
Determinant ≠ 0
Application:
Determinant ≠ 0
Function y1, y2, y3 are linearly independent.
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Community Answer
Consider the following statements about the linear dependence of the r...
Analysis:
To determine the linear dependence of the functions y1= 1, y2= x, and y3= x^2, we need to check if there exist constants c1, c2, and c3 not all zero such that c1*y1 + c2*y2 + c3*y3 = 0 for all x in the given interval.
For -1 ≤ x ≤ 0:
In this interval, y1 = 1, y2 = x, and y3 = x^2. We need to find constants c1, c2, and c3 such that c1*1 + c2*x + c3*x^2 = 0 for all x in this interval.
Since this equation is a quadratic, it can only be satisfied if c1 = c2 = c3 = 0. Therefore, y1, y2, and y3 are linearly independent in this interval.
For 0 ≤ x ≤ 1:
In this interval, we have the same functions y1, y2, and y3. We need to find constants c1, c2, and c3 such that c1*1 + c2*x + c3*x^2 = 0 for all x in this interval.
Since this equation is a quadratic, it can only be satisfied if c1 = c2 = c3 = 0. Therefore, y1, y2, and y3 are linearly independent in this interval.
Conclusion:
Therefore, the correct statement is that both I and III are true. In the given intervals, the functions y1, y2, and y3 are linearly independent.
To determine the linear dependence of the functions y1= 1, y2= x, and y3= x^2, we need to check if there exist constants c1, c2, and c3 not all zero such that c1*y1 + c2*y2 + c3*y3 = 0 for all x in the given interval.
For -1 ≤ x ≤ 0:
In this interval, y1 = 1, y2 = x, and y3 = x^2. We need to find constants c1, c2, and c3 such that c1*1 + c2*x + c3*x^2 = 0 for all x in this interval.
Since this equation is a quadratic, it can only be satisfied if c1 = c2 = c3 = 0. Therefore, y1, y2, and y3 are linearly independent in this interval.
For 0 ≤ x ≤ 1:
In this interval, we have the same functions y1, y2, and y3. We need to find constants c1, c2, and c3 such that c1*1 + c2*x + c3*x^2 = 0 for all x in this interval.
Since this equation is a quadratic, it can only be satisfied if c1 = c2 = c3 = 0. Therefore, y1, y2, and y3 are linearly independent in this interval.
Conclusion:
Therefore, the correct statement is that both I and III are true. In the given intervals, the functions y1, y2, and y3 are linearly independent.
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Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer?
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Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer?.
Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
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ample number of questions to practice Consider the following statements about the linear dependence of the real valued functions y1= 1, y2= x and y3= x2, over the field of real numbers.I. y1, y2and y3are linearly independent on – 1 ≤ x ≤ 0II. y1, y2and y3are linearly dependent on 0 ≤ x ≤ 1III. y1, y2and y3are linearly independent on 0 ≤ x ≤ 1IV. y1, y2and y3are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct?a)Both I and II are trueb)Both I and III are truec)Both II and IV are trued)Both III and IV are trueCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
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