In: Advanced Math

Let P^{N} denote the vector space of all polynomials of
degree N or less, with real coefficients. Let the linear
transformation: T: P^{3} --> P^{1} be the second
derivative. Is T onto? Explain. Is T one-to-one? What is the Kernel
of T? Find the standard matrix A for the linear transformation T.
Let B= {x+1 , x-1 , x^{2}+x , x^{3}+x^{2} }
be a basis for P^{3} ; and

F={ x+2 , x-3 } be a basis for P^{1} . Find A
_{F<--B} ( the matrix for T relative to the bases B and
F).

Let P denote the vector space of all polynomials with real
coefficients and Pn be the set of all polynomials in p
with degree <= n.
a) Show that Pn is a vector subspace of P.
b) Show that {1,x,x2,...,xn} is a basis
for Pn.

Let P2 be the vector space of all polynomials of
degree less than or equal to 2.
(i) Show that {x + 1, x2 + x, x − 1} is a basis for
P2.
(ii) Define a transformation L from P2 into
P2 by: L(f) = (xf)' . In other words,
L acts on the polynomial f(x) by first multiplying the function by
x, then differentiating. The result is another polynomial in
P2. Prove that L is a linear transformation....

Let
V be the space of polynomials with real coefficients of degree at
most n, and let D be the differentiation operator. Find all
eigenvectors of D on V.

3. We let ??(?) denote the set of all polynomials of degree at
most n with real coefficients.
Let ? = {? + ??3 |?, ? ??? ???? ???????}. Prove that T is a
vector space using standard addition and scalar multiplication of
polynomials in ?3(?).

Let V be the 3-dimensional vector space of all polynomials of
order less than or equal to 2 with real coeﬃcients.
(a) Show that the function B: V ×V →R given by B(f,g) = f(−1)g(−1)
+ f(0)g(0) + f(1)g(1) is an inner product and write out its Gram
matrix with respect to the basis (1,t,t2).
DO NOT COPY YOUR SOLUTION FROM OTHER SOLUTIONS

Verify all axioms that show that the set of second degree
polynomials is a vector space. What is the Rank?
P2 = {p(x)P | p(x) = ax^2 + bx + c where a,b,c E
R}

Problem 6. Let Pd (2, C) denote the vector space of
C-polynomials in two variables, of degree ≤ d.
Define a linear map S : P2(2, C) → P2(C) by S(p) := p(z, z)
(where z is a variable for the polynomials in P2(C)).
(a) Prove that S is surjective and that Skew2(2, C) ⊂
ker(S).
(b) Give an example of a polynomial in ker(S) \ Skew2(2, C).
Hence write down a basis for ker(S).

S_3 is the vector space of polynomials degree <= 3. V is a
subspace of poly's s(t) so that s(0) = s(1) = 0. The inner product
for two poly. s(t) and f(t) is def.: (s,f) = ([integral from 0 to
1] s(t)f(t)dt). I would like guidance finding (1) an orthogonal
basis for V and (2) the projection for s(t) = 1 - t + 2t^2. Thank
you!

4. Whether P3 or the space of the polynomials of degree less
than or equal to 3 and consider T: P3 → P3, given by the derivation
T(f) = f' . For example, T (−3x 2 + 5x - 10) = −6x + 5.
(a) Prove that T is a linear transformation.
(b) Determine ker (T).
(c) Is the T transformation injective? Justify that.
(d) The polynomial g (x) = 3x^2 + 1 belongs to the image?
Justify that.

1. Show that the set of all polynomials of deg=2 is not a vector
space over reals.
can this be fixed, can we have a set of polynomials that is a
vector space over reals?
2. Show that the set of 2x2 matrices with m_22 = 1 is not a
vector space over reals.
3. Show that the set of infinitely-differentiable real functions
is a a vector space under pointwise function addition, and
pointwise scalar multiplication as defined in class,...

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