Answered: 2. (a) Show that the list {p o(x), p… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 31E

See similar textbooks

Related questions

Question

2. (a) Show that the list {p o(x), p 1(x), p 2(x),...,p n(x),...,} where
Po(x) = 1, P₁(x) = 1 + x,
pj(x) = 1 + x + ·
+x²,
is a basis of the vector space F[x] of all polynomials with coeficients in F.
(b) What are the coordinates of the vector xn relative to the basis (po(x), p₁(x),..., pj(x), ...)?

Expert Solution

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Let P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let = 2+x+x², 2+x+x², 2+x², = 11 + 3x + 6x² choose PI(T) P2(x) P3(x) = P4(x) and let C = {p1(x), p2(x), P3(x), P4(x)}. a. Use coordinate representations with respect to the basis B = {1, 2, ², ³} to determine whether the set C forms a basis for P.. = c. The dimension of span(C) is b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. {}
{ƒf(x) = P₂[x] |ƒ'(−8) =ƒ(1)} where P₂[x] is the vector space of polynomials in x with degree at most 2. You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². p(x) = ,q(x) = =
2. (a) Show that the list {p o(x),p 1(x), p 2(x),... ,P n(x),...,} where Po(x) = 1, p1(x) = 1+x, P;(x) = 1+x + + x³, •..) is a basis of the vector space F[x] of all polynomials with coeficients in F. (b) What are the coordinates of the vector x" relative to the basis (po(x),P1(x),..., P; (x),...)?
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Let p(x) = 3 - 2x and q(x) = 1 + x + x2 (p(x), q(x)) is the inner product on the vector space P2 [ 0, 1]. Compute (a) <p(x), q(x)> (b) llp(x) II (c) d(p(x), q(x))
Show that the set of all second order polynomials, f(x) = ax² + bx+c, is a vector space.
(d) Show that the polynomials P₁ = 3 + 2x + x², P₂ = 2x+3x², p3= −2+x-2x² form a basis of the vector space P₂ of polynomials of degree at most 2 and find the coordinates of 1 + 2x - x² in this basis. Show your working.
Let P₂ be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial p(x) = x+x²-2 relative to the basis B = {1, x -2, x²-x}.

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS