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Hello. This is an exercise is from Hoffman, Linear Algebra, chapter 7.4; but it has no solution,...

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Advance Math

Hello.

This is an exercise is from Hoffman, Linear Algebra, chapter7.4; but it has no solution, Can you help me to understand how tosolve it? I have just a very general idea of how to solve it and Iam afraid that, if the degree of the polynomial is changed, I mayfail the solution.

4. Construct a linear operator T with minimal polynomial x^2 (x- 1)^2 and characteristic
polynomial x^3(x-1)^4. Describe the primary decomposition of thevector
space under T and find the projections on the primary components.Find a basis
in which the matrix of T is in Jordan form. Also find an explicitdirect sum decomposition
of the space into T-cyclic subspaces as in Theorem 3 and give theinvariant
factors.

Best Regards.

Answer & Explanation Solved by verified expert
4.2 Ratings (907 Votes)
Since we have to construct and in fact the Vectorspace on which is defined is notgiven Hence we can simply consider we choose dimension because givenCharacteristic polynomial is of degree we can choosematrix of with respect tostandard basis of to be a Jordan normalform Further by given Characteristic    See Answer
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