1.Q,1 find the coordinates of the point P which divides line segment QR in the ratio m : n in the following examples .
1. Q( -5, 8) R(4, -4) m : n = 2 : 1
2. Q (-2, 7) R(-2, -5) m : n = 1 :3. Please explain it
Answers
Step-by-step explanation:
Given a field FF, an integer n\geq 1n≥1, and a matrix A\in M_n(F)A∈M
n
(F), are there polynomials f,g\in F[X]f,g∈F[X], with ff monic of degree nn, such that AA is similar to g(C_f)g(C
f
), where C_fC
f
is the companion matrix of ff? For infinite fields the answer is easily seen to positive, so we concentrate on finite fields. In this case we give an affirmative answer, provided |F|\geq n-2∣F∣≥n−2. Moreover, for any finite field FF, with |F|=m∣F∣=m, we construct a matrix A\in M_{m+3}(F)A∈M
m+3
(F) that is not similar to any matrix of the form g(C_f)g(C
f
). Of use above, but also of independent interest, is a constructive procedure to determine the similarity type of any given matrix g(C_f)g(C
f
) purely in terms of ff and gg, without resorting to polynomial roots in FF or in any extension thereof. This, in turn, yields an algorithm that, given gg and the invariant factors of any AA, returns the elementary divisors of g(A)g(A). It is a rational procedure, as opposed to the classical method that uses the Jordan decomposition of AA to find that of g(A)g(A). Finally, extending prior results by the authors, we show that for an integrally closed ring RR with field of fractions FF and companion matrices C,DC,D the subalgebra R< C,D>R<C,D> of M_n(R)M
n
(R) is a free RR-module of rank n+(n-m)(n-1)n+(n−m)(n−1), where mm is the degree of \gcd (f,g)\in F[X]gcd(f,g)∈F[X], and a presentation for R< C,D>R<C,D> is given in terms of CC and DD. A counterexample is furnished to show that R< C,D>R<C,D> need not be a free RR-module if RR is not integrally closed. The preceding information is used to study M_n(R)M
n
(R), and others, as R[X]R[X]-modules.