Use induction to prove
Let f(x) be a polynomial of degree n in Pn(R). Prove that for
any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that
g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the
nth derivative of f(x).
a. Prove that for any vector space, if an inverse exists, then
it must be unique.
b. Prove that the additive inverse of the additive inverse will
be the original vector.
c. Prove that the only way for the magnitude of a vector to be
zero is if in fact the vector is the zero vector.
Prove the following:
Let f(x) be a polynomial in R[x] of positive degree n.
1. The polynomial f(x) factors in R[x] as the product of
polynomials of degree
1 or 2.
2. The polynomial f(x) has n roots in C (counting multiplicity).
In particular,
there are non-negative integers r and s satisfying r+2s = n such
that
f(x) has r real roots and s pairs of non-real conjugate complex
numbers as
roots.
3. The polynomial f(x) factors in C[x] as...
(Lower bound for searching algorithms) Prove: any
comparison-based searching algorithm on a set of n elements takes
time Ω(log n) in the worst case. (Hint: you may want to read
Section 8.1 of the textbook for related terminologies and
techniques.)
More world leaders have an economics degree than any other type
of degree. Discuss how the study of economics has helped you
understand the world around you and how the knowledge of economics
can be used to better the world. Also discuss how, with your
knowledge of economics, you can profit in order to help better your
life, your family’s life, your very close friends, or your
community.
we have the question that 'if F is a field with char 0, prove
that prime subfield of F is isomorphic to the field of Q'. I
already figure out the answer. BUT from the question I have other
question have risen in my brain. 1. what are the official
definition of the kernel of a map and the characteristics of a
field? 2. what is the link between the kernel and the char? 3. are
they equivalent in some...