Question

Let the set W be: all polynomials in P_{3} satisfying
that p(-t)=p(t),

Question: Is W a vector space or not?

If yes, find a basis and dimension

Answer #1

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...

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

Question 4. Consider the following subsets of the vector space
P3 of polynomials of degree 3 or less: S = {p(x) : p(1) = 0} and T
= {q(x) : q(0) = 1} Determine if these subsets are vectors spaces
with the standard operations for polynomials

Consider W as the set of all skew-symmetric matrices of size
3×3. Is it a vector space? If yes, then ﬁnd its dimension and a
basis.

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.

Let F
be a subfield of a field K satisfying the condition that the
dimension of K as a vector
space
over F is finite and equal to r. Let V be a vector space of finite
dimension n > 0 over K. Find
the
dimension of V as a vector space over F

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 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.

1. Let T be a linear transformation from vector spaces
V to W.
a. Suppose that U is a subspace of V,
and let T(U) be the set of all vectors w in W
such that T(v) = w for some v in V. Show that
T(U) is a subspace of W.
b. Suppose that dimension of U is n. Show that
the dimension of T(U) is less than or equal to
n.

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