Question

Consider P_{3} = {a + bx + cx^{2} +
dx^{3} |a,b,c,d ∈ R}, the set of polynomials of degree at
most 3. Let p(x) be an arbitrary element in P_{3}.

(a) Show P_{3} is a vector space.

(b) Find a basis and the dimension of P_{3}.

(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 P_{3}.

(e) Find a basis for the subspace of P_{3} consisting of
the polynomials with p(1) = 0.

Answer #1

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

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.

In this question we denote by P2(R) the set of functions {ax2 +
bx + c : a, b, c ∈ R}, which is a vector space under the usual
addition and scalar multiplication of functions. Let p1, p2, p3 ∈
P2(R) be given by p1(x) = 1, p2(x) = x + 2x 2 , and p3(x) = αx + 4x
2 . a) Find the condition on α ∈ R that ensures that {p1,
p2, p3} is a 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.

Define T : P2 → R3 via T(a+bx+cx2) = (a+c,c,b−c), and let B =
{1,x,x2} and D ={(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
(a) Find MDB(T) and show that it is invertible.
(b) Use the fact that MBD(T−1) = (MDB(T))−1 to find T−1. Hint: A
linear transformation is completely determined by its action on any
spanning set and hence on any basis.

Prove that the singleton set {0} is a vector subspace of the space
P4(R) of all polynomials of degree at most 3 with real
coefficients.

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

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

Let F be a ﬁeld (for instance R or C), and let P2(F) be the set
of polynomials of degree ≤ 2 with coeﬃcients in F, i.e.,
P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}.
Prove that P2(F) is a vector space over F with sum ⊕ and scalar
multiplication deﬁned as follows:
(a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x +
(a2 + b2)x^2
λ (b0 +...

We have learned that we can consider spaces of matrices,
polynomials or functions as vector spaces. For the following
examples, use the definition of subspace to determine whether the
set in question is a subspace or not (for the given vector space),
and why.
1. The set M1 of 2×2 matrices with real entries such that all
entries of their diagonal are equal. That is, all 2 × 2 matrices of
the form: A = a b c a
2....

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