Question

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.

Answer #1

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 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 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?

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.

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 Poly3(x) = polynomials in x of degree at most 2. They form a
3- dimensional space. Express the operator Q(p) = xp' + p'' .
as a matrix (i) in basis {1, x, x^2 }, (ii) in basis {1, x,
1+x^2 } .
Here, where p(x) represents a polynomial, p’ is its derivative,
and p’’ its second derivative.

Let P and Q be polynomials of degree at least one and let a be
the lead coefficient of P and b be the lead coefficient of Q. Prove
that
lim as n approaches infinity of P(n)/Q(n) = 0 deg Q > deg
P
a/b deg Q = deg P
infinity deg Q < deg P

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.

Determine if the given set V is a subspace of the vector space
W, where
a) V={polynomials of degree at most n with p(0)=0} and W=
{polynomials of degree at most n}
b) V={all diagonal n x n matrices with real entries} and W=all n
x n matrices with real entries
*Can you please show each step and little bit of an explanation
on how you got the answer, struggling to learn this concept?*

Consider P3 = {a + bx + cx2 +
dx3 |a,b,c,d ∈ R}, the set of polynomials of degree at
most 3. Let p(x) be an arbitrary element in P3.
(a) Show P3 is a vector space.
(b) Find a basis and the dimension of P3.
(c) Why is the set of polynomials of degree exactly 3 not a
vector space?
(d) Find a basis for the set of polynomials satisfying p′′(x) =
0, a subspace of P3.
(e) Find...

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