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Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa b) then which is false? (a) (b) T ^ - 1 = [[1, 0], [- 1, 1]] with respect to standard basis. (c) 3 a trivial subspace of P, (R) which is equal to eigen space of T. (d) y-axis in two dimensional plane will be eigen?
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Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa ...
False Statement: (d) y-axis in two dimensional plane will be eigen?
To determine whether the statement is true or false, we need to understand what eigenvalues and eigenvectors are and how they relate to linear transformations.
Eigenvalues and Eigenvectors
In linear algebra, an eigenvector of a linear transformation is a non-zero vector that is only scaled by the transformation. In other words, when a linear transformation is applied to an eigenvector, the resulting vector is a scalar multiple of the original eigenvector.
Formally, let T be a linear transformation from a vector space V to itself. A non-zero vector v in V is called an eigenvector of T if there exists a scalar λ such that T(v) = λv. The scalar λ is called the eigenvalue corresponding to the eigenvector v.
Linear Transformation T
The given linear transformation T maps a vector (a, bx) in R^2 to the vector (aa, b). Let's apply this transformation to the standard basis vectors (1, 0) and (0, 1) to determine the matrix representation of T.
T(1, 0) = (11, 0) = 11(1, 0)
T(0, 1) = (00, 1) = 0(0, 1)
From these results, we can see that the standard basis vectors (1, 0) and (0, 1) are not eigenvectors of T since the resulting vectors are not scalar multiples of the original vectors.
Eigenspace of T
The eigenspace of a linear transformation T is the set of all eigenvectors corresponding to a particular eigenvalue. In this case, we need to determine the eigenspace of T for a given eigenvalue.
Let's take an arbitrary vector (a, b) and apply T to it:
T(a, b) = (aa, b)
To find the eigenvectors, we set T(a, b) equal to λ(a, b) and solve for λ, a, and b:
(aa, b) = λ(a, b)
aa = λa
b = λb
From the second equation, we can see that b must be equal to λb for any eigenvalue λ. This implies that b must be 0, meaning the y-coordinate of any eigenvector is 0. Therefore, the eigenspace of T is the x-axis, not the y-axis.
Conclusion
Based on the analysis above, the false statement is (d) "y-axis in two-dimensional plane will be eigen." The eigenspace of the given linear transformation T is the x-axis, not the y-axis.
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Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa b) then which is false? (a) (b) T ^ - 1 = [[1, 0], [- 1, 1]] with respect to standard basis. (c) 3 a trivial subspace of P, (R) which is equal to eigen space of T. (d) y-axis in two dimensional plane will be eigen?
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Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa b) then which is false? (a) (b) T ^ - 1 = [[1, 0], [- 1, 1]] with respect to standard basis. (c) 3 a trivial subspace of P, (R) which is equal to eigen space of T. (d) y-axis in two dimensional plane will be eigen? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa b) then which is false? (a) (b) T ^ - 1 = [[1, 0], [- 1, 1]] with respect to standard basis. (c) 3 a trivial subspace of P, (R) which is equal to eigen space of T. (d) y-axis in two dimensional plane will be eigen? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let \ P / (R) -> R ^ 2 be the linear transformation T(a bx) = (aa b) then which is false? (a) (b) T ^ - 1 = [[1, 0], [- 1, 1]] with respect to standard basis. (c) 3 a trivial subspace of P, (R) which is equal to eigen space of T. (d) y-axis in two dimensional plane will be eigen?.
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