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Consider the following statement 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
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
Q. 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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Consider the following statement about the linear dependence of the re...
y1 = 1, y2 = x, y3 = x2
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Consider the following statement about the linear dependence of the re...
Understanding Linear Dependence and Independence
Linear dependence and independence of functions relate to whether a linear combination of these functions can equal zero without all coefficients being zero. The functions in question are:
- y1 = 1
- y2 = x
- y3 = x²
Analysis on the Interval -1 < x="" />< />
- I. Linear Independence:
- For -1 < x="" />< 0,="" the="" functions="" y1,="" y2,="" and="" y3="" are="" linearly="" independent.="" />
- No constants a, b, c (not all zero) satisfy the equation:
a(1) + b(x) + c(x²) = 0 for all x in this interval.
- Hence, statement I is true.
- IV. Linear Dependence:
- In the same interval (-1 < x="" />< 0),="" the="" functions="" cannot="" be="" expressed="" as="" a="" non-trivial="" linear="" combination="" that="" equals="" zero.="" />
- Therefore, statement IV is false.
Analysis on the Interval 0 < x="" />< />
- II. Linear Dependence:
- For 0 < x="" />< 1,="" y1,="" y2,="" and="" y3="" are="" linearly="" dependent.="" />
- Specifically, we can find a non-trivial relation, e.g., c1(1) + c2(x) + c3(x²) = 0, for suitable coefficients.
- Hence, statement II is true.
- III. Linear Independence:
- Conversely, on this interval, y1, y2, and y3 cannot be linearly independent due to the dependence established in statement II.
- Thus, statement III is false.
Conclusion
Based on the analysis:
- Statements I and II are true.
- Statements III and IV are false.
Therefore, the correct answer is option b): Both I and III are true.
Linear dependence and independence of functions relate to whether a linear combination of these functions can equal zero without all coefficients being zero. The functions in question are:
- y1 = 1
- y2 = x
- y3 = x²
Analysis on the Interval -1 < x="" />< />
- I. Linear Independence:
- For -1 < x="" />< 0,="" the="" functions="" y1,="" y2,="" and="" y3="" are="" linearly="" independent.="" />
- No constants a, b, c (not all zero) satisfy the equation:
a(1) + b(x) + c(x²) = 0 for all x in this interval.
- Hence, statement I is true.
- IV. Linear Dependence:
- In the same interval (-1 < x="" />< 0),="" the="" functions="" cannot="" be="" expressed="" as="" a="" non-trivial="" linear="" combination="" that="" equals="" zero.="" />
- Therefore, statement IV is false.
Analysis on the Interval 0 < x="" />< />
- II. Linear Dependence:
- For 0 < x="" />< 1,="" y1,="" y2,="" and="" y3="" are="" linearly="" dependent.="" />
- Specifically, we can find a non-trivial relation, e.g., c1(1) + c2(x) + c3(x²) = 0, for suitable coefficients.
- Hence, statement II is true.
- III. Linear Independence:
- Conversely, on this interval, y1, y2, and y3 cannot be linearly independent due to the dependence established in statement II.
- Thus, statement III is false.
Conclusion
Based on the analysis:
- Statements I and II are true.
- Statements III and IV are false.
Therefore, the correct answer is option b): Both I and III are true.
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Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE.
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Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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?, a detailed solution for Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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? has been provided alongside types of Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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? theory, EduRev gives you an
ample number of questions to practice Consider the following statement 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 <0II. y1 , y2 and y3 are linearly dependent on 0 < x< 1III. y1, y2 and y3 are linearly independent on 0 < x< 1IV. y1, y2 and y3 are linearly dependent on - 1 < x < 0Q.Which 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 GATE tests.
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