Question

Let V be the set of polynomials of the form ax + (a^2)(x^2), for all real numbers a. Is V a subspace of P?

Answer #1

Let H be the set of all polynomials of the form p(t) = at2 where
a ∈ R with a ≥ 0. Determine if H is a subspace of P2. Justify your
answers.

5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that
p(-1)=0.
(a) Prove that S is a subspace of the vector space of all
polynomials.
(b) Find a basis for S.
(c) What is the dimension of S?
6.
Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2
=(1,2,-6,1),
?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2
=(3,1,2,-2). Prove that V=W.

Let ℙn be the set of real polynomials of degree at most n, and
write p′ for the derivative of p. Show that
S={p∈ℙ9:p(2)=−1p′(2)}
is a subspace of ℙ9.

Let R[x] be the set of all polynomials (in the variable x) with
real coefficients. Show that this is a ring under ordinary addition
and multiplication of polynomials.
What are the units of R[x] ?
I need a legible, detailed explaination

If V is a vector space of polynomials of degree n with real
numbers as coefficients, over R, and W is generated by
the polynomials (x 3 + 2x 2 − 2x + 1, x3 + 3x 2 − x + 4, 2x 3 +
x 2 − 7x − 7),
then is W a subspace of V , and if so, determine its basis.

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

Let P2 denote the vector space of polynomials in x with real
coefficients having degree at most 2. Let W be a subspace of P2
given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper
subspace of P2.

Let P be the vector space of all polynomials in x with real
coefficients. Does P have a basis? Prove your answer.

4) Let V be the subspace of C[a,b] spanned by e^x ; e^−x and let
Ax be the anti-differentiation operator that also holds the
constant of integration to be zero. Example: Ax(2e^−x ) = −2e^−x .
Find the matrix that represents Ax() in the standard ordered basis
e^x ; e^−x and call that matrix A. Find the matrix that represents
Ax() in the non-standard ordered basis cosh(x); sinh(x) and call
that matrix B. And write matrices A and B as...

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