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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. 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?
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Consider the linear transformation T : R7----> R7 defined by T (x1, x2...
Solution:
In order to analyze the given linear transformation T : R7 ----> R7, we will consider each option and determine its validity.
a) The determinant of T is different from 1:
To find the determinant of T, we need to construct the matrix representation of T. Let's denote the standard basis of R7 as e1, e2, ..., e7, where each ei has a 1 in the i-th position and 0s elsewhere.
The matrix representation of T with respect to the standard basis is:
[T] = [0 0 0 0 0 0 1
0 0 0 0 0 1 0
0 0 0 0 1 0 0
0 0 0 1 0 0 0
0 0 1 0 0 0 0
0 1 0 0 0 0 0
1 0 0 0 0 0 0]
Calculating the determinant of T, we find that det(T) = (-1)^15 * det([T]) = -1 * 1 = -1.
Therefore, the determinant of T is not equal to 1. Hence, option (a) is true.
b) There is no basis of R7 with respect to which T is diagonalizable:
For a linear transformation to be diagonalizable, it must have a basis of eigenvectors. Let's find the eigenvalues and eigenvectors of T.
The characteristic equation of T is given by |T - λI| = 0, where λ is the eigenvalue and I is the identity matrix.
Solving the characteristic equation, we get λ^7 - 1 = 0.
The eigenvalues of T are the 7th roots of unity, which are 1, e^(2πi/7), e^(4πi/7), e^(6πi/7), e^(8πi/7), e^(10πi/7), and e^(12πi/7).
Since T has distinct eigenvalues, it is diagonalizable. Therefore, option (b) is false.
c) T^7 = I:
To find T^7, we can calculate T^7(x) as T(T(T(T(T(T(T(x)))))).
(T(x))^7 = ((x7, x6, x5, x4, x3, x2, x1))^7
= (x1, x2, x3, x4, x5, x6, x7)
= x.
Therefore, T^7(x) = x for any vector x in R7.
Since T^7 is the identity transformation, option (c) is true.
d) The smallest n such that T^n = I is even:
From the previous calculation, we found that T^7 = I. Therefore, the smallest n such that T^n = I is 7, which is an odd number.
Hence, option (d) is false.
Summary:
From the analysis, we can conclude that the correct statements are:
a) The determinant of T is different from 1.
c) T^7 =
In order to analyze the given linear transformation T : R7 ----> R7, we will consider each option and determine its validity.
a) The determinant of T is different from 1:
To find the determinant of T, we need to construct the matrix representation of T. Let's denote the standard basis of R7 as e1, e2, ..., e7, where each ei has a 1 in the i-th position and 0s elsewhere.
The matrix representation of T with respect to the standard basis is:
[T] = [0 0 0 0 0 0 1
0 0 0 0 0 1 0
0 0 0 0 1 0 0
0 0 0 1 0 0 0
0 0 1 0 0 0 0
0 1 0 0 0 0 0
1 0 0 0 0 0 0]
Calculating the determinant of T, we find that det(T) = (-1)^15 * det([T]) = -1 * 1 = -1.
Therefore, the determinant of T is not equal to 1. Hence, option (a) is true.
b) There is no basis of R7 with respect to which T is diagonalizable:
For a linear transformation to be diagonalizable, it must have a basis of eigenvectors. Let's find the eigenvalues and eigenvectors of T.
The characteristic equation of T is given by |T - λI| = 0, where λ is the eigenvalue and I is the identity matrix.
Solving the characteristic equation, we get λ^7 - 1 = 0.
The eigenvalues of T are the 7th roots of unity, which are 1, e^(2πi/7), e^(4πi/7), e^(6πi/7), e^(8πi/7), e^(10πi/7), and e^(12πi/7).
Since T has distinct eigenvalues, it is diagonalizable. Therefore, option (b) is false.
c) T^7 = I:
To find T^7, we can calculate T^7(x) as T(T(T(T(T(T(T(x)))))).
(T(x))^7 = ((x7, x6, x5, x4, x3, x2, x1))^7
= (x1, x2, x3, x4, x5, x6, x7)
= x.
Therefore, T^7(x) = x for any vector x in R7.
Since T^7 is the identity transformation, option (c) is true.
d) The smallest n such that T^n = I is even:
From the previous calculation, we found that T^7 = I. Therefore, the smallest n such that T^n = I is 7, which is an odd number.
Hence, option (d) is false.
Summary:
From the analysis, we can conclude that the correct statements are:
a) The determinant of T is different from 1.
c) T^7 =
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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. 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?
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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. 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? 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----> R7 defined by T (x1, x2,., x6, x7) = (x7, x6,.,x2,x1) 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? 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----> R7 defined by T (x1, x2,., x6, x7) = (x7, x6,.,x2,x1) 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?.
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. 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? 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----> R7 defined by T (x1, x2,., x6, x7) = (x7, x6,.,x2,x1) 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? 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----> R7 defined by T (x1, x2,., x6, x7) = (x7, x6,.,x2,x1) 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?.
Solutions for 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. 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? in English & in Hindi are available as part of our courses for Mathematics.
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