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Let {v1, v2, ........,v16} be an ordered basis for V= C16. If T is a linear transformation on V defined by T(v) = vi+1 for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + ... + v16) Then,
  • a)
    T is singular with rational eigen values
  • b)
    T is singular but has no rational eigen values
  • c)
    T is regular (invertible) with rational eigen values
  • d)
    T is regular but has no rational eigen values
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a li...
Let B = {v1, v2, ........,v16} be an ordered basis for V= C16.
If T is a linear transformation on V defined by T(v) = vi+1 for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + .......+ v16).
Thus, the matrix of T relative to the ordered basis B = {v1, v2, ........,v16} is

Then Rank [TB]= 16
Hence, T is invertible and it has rational eigen values.
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Most Upvoted Answer
Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a li...
To determine the matrix representation of T with respect to the given basis, we need to find the images of the basis vectors under T.

T(v1) = v11 = 1⁡₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀
T(v2) = v21 = ₀₁₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀
T(v3) = v31 = ₀₀₁₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀
...
T(v16) = v161 = ₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₀₁

We can now construct the matrix representation of T by arranging the images of the basis vectors as columns:

| 1 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
| ₀ ₁ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
| ₀ ₀ ₁ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
...
| ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ |

Therefore, the matrix representation of T with respect to the given basis is:

| 1 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
| ₀ ₁ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
| ₀ ₀ ₁ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ |
...
| ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ |
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Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a linear transformation on V defined by T(v) = vi+1for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + ... + v16) Then,a)T is singular with rational eigen valuesb)T is singular but has no rational eigen valuesc)T is regular (invertible) with rational eigen valuesd)T is regular but has no rational eigen valuesCorrect answer is option 'C'. Can you explain this answer?
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Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a linear transformation on V defined by T(v) = vi+1for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + ... + v16) Then,a)T is singular with rational eigen valuesb)T is singular but has no rational eigen valuesc)T is regular (invertible) with rational eigen valuesd)T is regular but has no rational eigen valuesCorrect answer is option 'C'. 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 Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a linear transformation on V defined by T(v) = vi+1for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + ... + v16) Then,a)T is singular with rational eigen valuesb)T is singular but has no rational eigen valuesc)T is regular (invertible) with rational eigen valuesd)T is regular but has no rational eigen valuesCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let {v1, v2, ........,v16} be an ordered basis for V= C16.If T is a linear transformation on V defined by T(v) = vi+1for 1 ≤ i ≤ 15 and T(v16) = -(v1 + v2 + ... + v16) Then,a)T is singular with rational eigen valuesb)T is singular but has no rational eigen valuesc)T is regular (invertible) with rational eigen valuesd)T is regular but has no rational eigen valuesCorrect answer is option 'C'. Can you explain this answer?.
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