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Let $T:\mathbb R^7 \longrightarrow \mathbb R^7$ be the linear transformation given by $$ T(x_1, x_2, x_3, x_4 ,x_5, x_6, x_7) = (x_7, x_6, x_5 ,x_4 ,x_3 ,x_2 ,x_1) $$ Which of the following statements are true?

a. Determinant of $T$ is $1$

b. Basis of $\mathbb R^7$ w.r.t. $T$ is a diagonal matrix

c. $T^7 =I$

d. Smallest $n$ s.t. $T^n =I$ is even.

I am stuck on this problem and don't know where to begin. Can anyone help me please?

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    http://en.wikipedia.org/wiki/Permutation_matrix2012-12-17
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    It seems like you dont want to do anything and just grasp the answer here. Show some effort, try as much as you can and you will get hints from the community2012-12-17
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    At least you should try (c), which is pretty straightforward. Just compute $T(T(T(T(T(T(T(x_1,x_2,x_3,x_4,x_5,x_6,x_7)))))))$ and see if it is identical to $\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7\right)$.2012-12-17

2 Answers 2