Question

Apply Gram-Schmidt in L2[−1,1] to the list of functions 1,x,x2,x3. (You do not have to normalize.)

Answer #1

Consider interval [0,2] and an inner product with weight w(x)=1.
Starting with 1, x, x^2, x^3 and using Gram-Schmidt process, build
four polynomials of degree 0,1,2,3 orthogonal on [0,2]. You do not
have to normalize them (I think this simplifies calculations). Just
remember that if they are not normalized, Gram-Schmidt formula will
have denominators in the form <q_j,q_j>. (If you do normalize
them, then <q_j,q_j> = 1).

Applythe Gram-schmidt orthonormalizaion proceudre to the set
1,x2,x4 which is linearly independent on the
interval [1,2] to construct a mutually orthonormal system phi_1(x),
phi_2(x), phi_3(x)

1) Determine whether x3 is O(g(x)) for the following:
a. g(x) = x2 + x3
b. g(x) = x2 + x4
c. g(x) = x3 / 2 2)
Show that each of these pairs of functions are of the same
order:
a. 3x + 7, x
b. 2x2 + x - 7, x2

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 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1 + x2 + x3=1 and
xi >0 for i = 1,2,3. find the
distribution of X1 ? Find
E(X1).

**NUMBER THEORY**
Proof that the following polynomials do not have integer
roots.
a) x3 − x + 1
b) x3 + x2 − x + 1

**NUMBER THEORY**
Proof that the following polynomials do not have integer
roots.
a) x3 + x2 − x + 3
b) x5 − x2 + x − 3.

•List three variables (X1, X2, X3) you’d include in a Multiple
Regression Model in order to better predict an outcome (Y)
variable. For example, you might list three variables that could be
related to how long a person will live (Y). Or you might list three
variables that contribute to a successful restaurant. Your
Regression Model should have three variables that will act as
“predictors” (X1, X2, X3) of a “criterion” (Y’). Note that the
outcome or criterion variable (e.g....

Find the derivatives of each of the following functions. DO NOT
simplify your answers.
(a) f(x) = 103x (3x5+ x − 1)4
(b) g(x) = ln(x3 + x) /
x2 − 4
(c) h(x) = tan-1(xex)
(d) k(x) = sin(x)cos(x)

Let the vectors a and b be in
X =
Span{x1,x2,x3}.
Assume all vectors are in R^n for some positive integer n.
1. Show that 2a - b is in
X.
Let x4 be a vector in Rn that is not contained
in X.
2. Show b is a linear combination of
x1,x2,x3,x4.
Edit: I don't really know what you mean, "what does the question
repersent." This is word for word a homework problem I have for
linear algebra.

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