From a linear algebra course, I included definition 8.5 which is referenced in the question 8.C.7: Suppose V is a complex vector space. Suppose that PE£(V) is such that P2 =P. Prove thatthe characteristic polynomial of P is z"(z- 1)* where m= dim null P and k= dim range P.HINT: This problem has many moving parts,First, if P2 = P, what does that give you when you apply (8.5)? Second,use the previous decomposition to show that P can only have eigenvalues of zero and one.Third, remember that multiplicities must add-up to the dimension, and again use thedecomposition from the first step.8.5 V is the direct sum of null T dim V and range Tdim VSuppose T e L(V). Let n = dim V. ThenV = null T" range T".

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Answered: 8.C.7: Suppose V is a complex vector…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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From a linear algebra course, I included definition 8.5 which is referenced in the question

8.C.7: Suppose V is a complex vector space. Suppose that PE£(V) is such that P2 =P. Prove that
the characteristic polynomial of P is z"(z- 1)* where m= dim null P and k= dim range P.
HINT: This problem has many moving parts,
First, if P2 = P, what does that give you when you apply (8.5)? Second,
use the previous decomposition to show that P can only have eigenvalues of zero and one.
Third, remember that multiplicities must add-up to the dimension, and again use the
decomposition from the first step.
8.5 V is the direct sum of null T dim V and range Tdim V
Suppose T e L(V). Let n = dim V. Then
V = null T" range T".

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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