Prove that if two transformations are topologically conjugate and one one of them is topologically mixing, so the other is also mixing topologically.
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Prove that if two transformations are topologically conjugate and one
one of them is topologically mixing, so the other is also mixing
topologically.
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- Prove that (a) the zero transformation and (b) the identity transformation are linear transformations.Prove that if two transformations are topologically conjugate and one of them mixestopologically, then the other is also topologically mixed.If both t and s are self adjoint linear transformation on an inner product space V, then ts + st is self adjoint. It both t ands are skew-adjoint, then ts- st is skew-adjoint.
- A linear transformation T is said to be nilpotent if T" 0 for some n. Show that a nilpotent transformation is diagonalisable if and only if it is the 0 linear transformation.Prove that S + T and cT are linear transformations.polic ...the co.... 3. Prove that a general linear transformation maps circles to circles.
- 2. Does there exist a linear transformation T: R4 → R with nullspace X2 ker (T) = -(13-- = X3 X4 € R¹ : £1 = 3x2, x3 = x4 If yes, give an example of such a transformation. If not, give a proof that no such transfor- mation can exist.d) Prove that the composition of two surjective linear transformations is again a surjective linear transformation. e) Prove that if U and V are isomorphic then dim U = dim V, i.e their dimensions are the same. f) Prove that if the dimensions of U are the same as dimensions of V, then U and V must be isomorphic.7. Which of the following types of linear transformations are always diagonalisable? (a) The idempotent transformations, satisfying T² = T. %3D (b) The nilpotent transformations, having a power which is zero (c) The invertible linear transformations (d) The linear transformations whose minimal polynomial factors completely into distinct linear factors. (e) The self adjoint transformations on an inner product space.