In: Advanced Math

Let P_{2} be the vector space of all polynomials of
degree less than or equal to 2.

(i) Show that {x + 1, x^{2} + x, x − 1} is a basis for
P_{2}.

(ii) Define a transformation L from P_{2} into
P_{2} 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
P_{2}. Prove that L is a linear transformation.

(iii) Compute the matrix representation of the linear
transformation L above with respect to the basis for P_{2}
from the first part of this problem.

Consider the vector space P2 of all polynomials of degree less
than or equal to 2 i.e. P = p(x) = ax + bx + c | a,b,c €.R
Determine whether each of the parts a) and b) defines a subspace in
P2 ? Explain your answer. a) ( 10 pts. ) p(0) + p(1) = 1 b) ( 10
pts.) p(1) = − p(−1)

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 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 ....

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.

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}

Recall P2(t) is the set of polynomials of order less than or
equal to 2. Consider the the set of vectors in P2(t).
B={t^2,(t−1)^2,(t+1)^2}
(a) Show B is a basis for P2(t).
(b) If E={1,t,t^2}is the standard basis, calculate the change of
basis matrices PE→B and PB→E
(c) Given v= 2t^2−5t+ 3, find its components in B

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.

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
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 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.

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