Answered: Let P3 be the vector space of… | 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 22E

See similar textbooks

Related questions

Question

Let P3 be the vector space of polynomials of degree at most 3, with real coefficients. Consider the linear
transformation T: P3 → P3 given by
For example, T(x² + 1) = (x − 2)² + 1 = x² − 4x + 5.
(a) Compute the following polynomials:
T(1) =
T(p(x)) = p(x − 2).
T(x) =
T(x²) =
T(x³) =
T(x² + 2x) =
(b) Find the matrix of T with respect to the basis B = {1, x, x², x³}.
[T] B.B =

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Write the polynomial f(t) = af + bt +c as a linear combination of the polynomials p = (t –1), P2 =t-1, P3 = 1. [Thus, P, P, Ps span the space P2(t) of polynomials of degree < 2.]
Suppose that f(x) = x² + 1, g(x) = x² – x and h(x) = x – 1. Then the set of functions {f, g, h} is a basis for P2, the set of all polynomial functions of degree less than or equal to two. - 14x? – 2x – 2 as a linear combination of these basis functions: Express the function A(x) A = cif + c2g + c3h. Ci = C2 = C3 =
(b) Are the polynomials x + 5, – 5, 2x +1 linearly independent in P1 (R)? Explain.
Find an orthogonal change of variables, x = Py, that transforms the quadratic form q(x) = x² + x²+2x1x2 − 2x2x3 into a quadratic form no cross-product form. Write the new quadratic form.
Show that p(x)=5x − 3x2 polynomial in P2 vector space can be written as a linear combination of the set {3 + x2,1 − x, 1 + 3x − x2}.
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. {}
T: P2 P2 defined by T(a + bx + cx²) = a + b(x + 8) + b(x + 8)² Find the following images of the function for vectors p₁(x) = a₁ + b₁x + c₁x² and p₂(x) = a₂ + b₂x + c₂x2 in P₂ and the scalar k. (Give all answers in terms of a₁, b₁, C₁, 22, b₂, C₂ and c.) T(P₁) + T(P₂): = T(P1 + P₂) = cT (P1) = T(CP1) = Determine whether T is a linear transformation. 4 Olinear transformation O not a linear transformation
Let T:V->V be a linear operator and let p(x) be any polynomial. Then the kernel of p(T) is T-invariant.
2 Suppose that f(x) = x² + 1, g(x) = x² x and h(x) : х — Then the set of functions {f, g, h} is a basis for P2, the set of all = x – 1. polynomial functions of degree less than or equal to two. - 10x² + 5x + 3 as a linear Express the function A(x) combination of these basis functions: A = cif + c2g + c3h. C1 = C2 = C3
The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z + 4x²is -8- 8 in P₂. The standard basis polynomials for this space are {1, 2, z²}. represented by the vector The function F, defined by F(p(z)) = (x+3). takes the derivative of p(x) and then multiplies the result by (z + 3). a) Write the matrix M for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] M= b) The number 0 is an eigenvalue for this transformation. Draw three different non-zero polynomials in P₂ that are eigenvectors corresponding to λ = 0. Hint -6-5-4-3-2 Clear All Draw: Hint -2 Hint 4 -2 -3. c) The number 1 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are eigenvectors corresponding to λ = 1. Clear All Draw: -6 -5 -4 -3 0 5 4- 3 1 -1 -4 -5. -6+ 6 4 3- 2- 1 -1 -3 -4 -5 -6 d P(2), is a linear transformation from P₂ to P₂. It /^ d) The number 2…
3- (a) u=- 4 +31 +3, v=+2 + 4t-1, w = 21 -- 31+5; (b) u=- S - 21 + 3, v-4 - 31 + 4, w= 2r - 17 - 71 +9. Determine whether the following polynomials u, v, w in P(1) are linearly dependent or independent:
{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) = =

SEE MORE QUESTIONS

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS