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Consider the linear transformation T : R7----> R7 defined by T (x1, x2,......, x6, x7) = (x7, x6,..........,x2,x1)
Q. Which of the following statements is true.
Q. Which of the following statements is true.
- a)The determinant of T is different from 1
- b)There is no basis of R7 with respect to which T is diagonalisable
- c)T7 = I
- d)The smallest n such that Tn = I is even
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Consider the linear transformation T : R7----> R7defined byT (x1, x...
We are given that a linear transformation T : R7 —> R7 defined by
Let
be an ordered basis of R
then
So, the matrix corresponding to linear transformation T with respect to basis B is
Implies TB = I7x7
Now
= 1
Thus, determinant of T is 1.
Now,
T(x1, x2, . . . , x6, x7) = (x7, x6, . . . , x2, x1)
Therefore, ,T(x1, x2, . . . , x6, x7)= T(T(x1, x2, . . . , x6, x7)) = T(x
= (x1, x2, . . . , x6, x7)
Implies that T2 = 1
Hence, the smallest n such that Tn = I is 2 which is an even number.
Also ,T3(x1, x2, . . . , x6, x7)
= T2(T(x1, x2, . . . , x6, x7)
= (x7, x6, . . . , x2, x1)
= T(x1, x2, . . . , x6, x7)
Implies that T3 = T
Implies T6 = (T3)2 = T2 = I Therefore,T7(x1, x2, . . . , x6, x7)
= T6(x7, x6, . . . , x2, x1) = T(x1, x2, . . . , x6, x7)
Implies T7 = T ≠ I
Let
be an ordered basis of R7
then
So, the matrix corresponding to linear transformation T with respect to basis B isImplies TB = I7x7
Now
= 1Thus, determinant of T is 1.
Now,
T(x1, x2, . . . , x6, x7) = (x7, x6, . . . , x2, x1)
Therefore, ,T(x1, x2, . . . , x6, x7)= T(T(x1, x2, . . . , x6, x7)) = T(x
7
, x6, . . . , x2, x1)= (x1, x2, . . . , x6, x7)
Implies that T2 = 1
Hence, the smallest n such that Tn = I is 2 which is an even number.
Also ,T3(x1, x2, . . . , x6, x7)
= T2(T(x1, x2, . . . , x6, x7)
= (x7, x6, . . . , x2, x1)
= T(x1, x2, . . . , x6, x7)
Implies that T3 = T
Implies T6 = (T3)2 = T2 = I Therefore,T7(x1, x2, . . . , x6, x7)
= T6(x7, x6, . . . , x2, x1) = T(x1, x2, . . . , x6, x7)
Implies T7 = T ≠ I
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Consider the linear transformation T : R7----> R7defined byT (x1, x...
The question is incomplete as it does not provide the rest of the information about the linear transformation T. Please provide the missing information for further assistance.
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Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer?
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Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer?.
Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the linear transformation T : R7----> R7defined byT (x1, x2,......, x6, x7) =(x7,x6,..........,x2,x1)Q. Which of the following statements is true.a)The determinant of T is different from 1b)There is no basis of R7 with respect to which T is diagonalisablec)T7 = Id)The smallest n such that Tn = Iis evenCorrect answer is option 'D'. Can you explain this answer?.
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