In: Advanced Math

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.

Let PN denote the vector space of all polynomials of
degree N or less, with real coefficients. Let the linear
transformation: T: P3 --> P1 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 , x2+x , x3+x2 }
be a basis for P3 ; and
F={ x+2 , x-3 } be a basis for P1 ....

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.

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(?).

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!

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 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

Let V be the vector space of 2 × 2 real matrices and let P2 be
the vector space of polynomials of degree less than or equal to 2.
Write down a linear transformation T : V ? P2 with rank 2. You do
not need to prove that the function you write down is a linear
transformation, but you may want to check this yourself. You do,
however, need to prove that your transformation has rank 2.

Let Poly3(x) = polynomials in x of degree at most 2. They form a
3- dimensional space. Express the operator T(p) = p'
as a matrix (i) in basis {1, x, x 2 }, (ii) in basis {1, x, 1+x
2 } .

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.

Let V be a vector space and let U and W be subspaces of V . Show
that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V
.

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