Question

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.

Answer #1

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?

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.

Consider V = fp(t) 2 P2 : p0(1) = p(0)g, where P2 is a set
of
all polynomials of degree less than or equal to 2.
(1) Show that V is a subspace of P2
(2) Find a basis of V and the dimension of V

Let the set W be: all polynomials in P3 satisfying
that p(-t)=p(t),
Question: Is W a vector space or not?
If yes, find a basis and dimension

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.

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

Let W⊂ C1 be the subspace spanned by the two polynomials x1(t) =
1 and x2(t) =t^2. For the given function y(t)=1−t^2 decide whether
or not y(t) is an element of W. Furthermore, if y(t)∈W, determine
whether the set {y(t), x2(t)} is a spanning set for W.

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 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

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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