Question

Let T: P3→P3 be given by T(p(x)) =p(2x) and consider the bases B={1, x, x2, x3}and B′={1, x−1,(x−1)2,(x−1)3} of P3.

(a) Find the matrix of T relative to B.

(b) Find the change of basis matrix from B′ to B.

Answer #1

Let B = {(1, 2), (−1, −1)} and B' = {(−4, 1), (0, 2)} be bases
for R2, and let A = −1 2 1 0 be the matrix for T: R2 → R2 relative
to B. (a) Find the transition matrix P from B' to B. P =
(b) Use the matrices P and A to find [v]B and [T(v)]B , where
[v]B' = [−3 1]T. [v]B = [T(v)]B =
(c) Find P inverse−1 and A' (the matrix for...

Let B = {(1, 3), (?2, ?2)} and B' = {(?12, 0), (?4, 4)} be bases
for R2, and
let A =
3
2
0
4
be the matrix for T: R2 ? R2 relative to B.
(a) Find the transition matrix P from B' to B. P =
(b) Use the matrices P and A to find [v]B and [T(v)]B, where
[v]B' = [1 ?5]T. [v]B = [T(v)]B =
(c) Find P?1 and A' (the matrix for T relative...

Find the matrix operator T: P3 --> P2 where T [= T(a + bx +
cx^2 + dx^3) = b + 2cx + 3dx^2 with respect to bases B = {1, x,
x^2, x^3} and C = {1, x, 2x^2, -1}.

Find the derivative of the following functions
(a) f(x) = ln(√x3 −2x)
(b) g(x) =√x2 + 3 x3 −5x + 1
.

Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...

Consider P3 = {a + bx + cx2 +
dx3 |a,b,c,d ∈ R}, the set of polynomials of degree at
most 3. Let p(x) be an arbitrary element in P3.
(a) Show P3 is a vector space.
(b) Find a basis and the dimension of P3.
(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 P3.
(e) Find...

Consider the following two ordered bases of R^2:
B={〈1,−1〉,〈2,−1〉}
C={〈1,1〉,〈1,2〉}.
Find the change of coordinates matrix from the basis B to the
basis C.
PC←B=?
Find the change of coordinates matrix from the basis C to the
basis B.
PB←C=?

Define T:R^2 -> R^2 by T(x,y) = (x+3y, -x+5y) and let B' =
{(3,1)^T, (1,1)^T}. Find the matrix A' for T relative to the basis
B'.

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

Let A equal the 2x2 matrix:
[1 -2]
[2 -1]
and let T=LA R2->R2. (Notice
that this means T(x,y)=(x-2y,2x-y), and that the matrix
representation of T with respect to the standard basis is A.)
a. Find the matrix representation [T]BB
where B={(1,1),(-1,1)}
b. Find an invertible 2x2 matrix Q so that [T]B =
Q-1AQ

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