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 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 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 F be a field.
(a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) =
2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple.
That is, a^2 + b^2 = c^2. Use this fact to explain how to generate
right triangles with integer side lengths.
(b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy
− y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...
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 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 ....
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!