Question

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given second-degree polynomials form an orthonormal set, and if not, then apply the Gram-Schmidt orthonormalization process to form an orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in both answer blanks.)

{ square root 3 (x2−1), square root 3 (x2 + x + 2)}

u1 =

u2 =

Answer #1

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the
polynomials form an orthonormal set, and if not, apply the
Gram-Schmidt orthonormalization process to form an orthonormal set.
(If the set is orthonormal, enter ORTHONORMAL in both answer
blanks.)
{−2 + x2, −2 + x}
u1=
u2=

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given
second-degree polynomials form an orthonormal set, and if not, then
apply the Gram-Schmidt orthonormalization process to form an
orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in
both answer blanks.) { 3 (x2−1), 3 (x2 + x + 2)}
u1 =
u2 =

Let F be a ﬁeld (for instance R or C), and let P2(F) be the set
of polynomials of degree ≤ 2 with coeﬃcients in F, i.e.,
P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}.
Prove that P2(F) is a vector space over F with sum ⊕ and scalar
multiplication deﬁned as follows:
(a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x +
(a2 + b2)x^2
λ (b0 +...

Find two linearly independent solutions of
2x2y′′−xy′+(−2x+1)y=0,x>0 of the form
y1=xr1(1+a1x+a2x2+a3x3+⋯) y2=xr2(1+b1x+b2x2+b3x3+⋯) where
r1>r2.
Enter r1= a1= a2= a3= r2= b1= b2= b3=
Note: You can earn partial credit on this problem.

problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +
anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials
where the constant coefficient is a multiple of 5. You can assume
that I is an ideal of Z[x]. a. What is the simplest form of an
element in the quotient ring z[x] / I? b. Explicitly give the
elements in Z[x] / I. c. Prove that I is not a...

Let f(x)=−6, g(x)=−4x+4 and h(x)=−9x2. Consider the inner
product 〈p,q〉=∫40p(x)q(x)dx in the vector space C0[0,4] of
continuous functions on the domain [0,4]. Use the Gram-Schmidt
process to determine an orthonormal basis for the subspace of
C0[0,4] spanned by the functions f(x), g(x), and h(x).

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