Let V be the vector space P(t) of all polynomials in t over R, with inner product defined by (f(t), g(t)) = So f(t)g(t) dt. Let W be the subspace P2(t) of V. (a) Show that B = {1,2t – 1, 6t2 – 6t + 1} is an orthogonal basis for W. (b) Find the projection of f(t) = t3 onto W.

Let V be the vector space P(t) of all polynomials in t over R, with inner product defined by (f(t), g(t)) = So f(t)g(t) dt. Let W be the subspace P2(t) of V. (a) Show that B = {1,2t – 1, 6t2 – 6t + 1} is an orthogonal basis for W. (b) Find the projection of f(t) = t3 onto W.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let W = Span(v₁, V2, V3) where 1 --0--0--0 1 -1 3 Note that V₁, V2, V3 is an orthogonal basis of W.…

A: Let , where:.Let .We know that, if is an orthogonal basis of a vector subspace , than orthogonal…

Q: A ball is drawn randomly from a jar that contains 5 red balls, 7 white balls, and 4 yellow balls.…

A: From the given information, there are 5 red balls, 7 red balls and 4 yellow balls. The total number…

Q: Let V R2[x] be the real vector space of polynomials of degree at most 2. Let {fo, f1, f2} with fo(x)…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: 2 Let R¹ be the vector space over R with standard addition and scalar multiplication. Let X = {(x,…

A: X=x,y,z,w∈ℝ4: x-2y-3y-4z=0

Q: Let P₂(R) be the vector space of all real polynomials of at most degrees 2 with the inner product…

A: Given that

Q: Let V be the vector space of functions spanned by Nπ B = {b₁ = 5 sin x + 7 cos x, b₂ = 3 sin x + 9…

Q: Let V = P² be the vector space of polynomials of degree at most 2, and let B be the basis {f1, f2,…

Q: vectors! the following Consider X2= x₁ =[ ₁ ] , x 2 = [ 2 ] ₁ x 3 = 12 D C X₁, X ₁, X3 6 X3= 16 -5 D…

Q: Let & denote the vector space of all functions &: R&R under the usual operations of function…

A: B=sint, cost is a basis of H

Q: Let be the vector space of polynomials of degree less than or equal to one with real coefficients…

A: The given vector space is the set of all polynomials of degree less than or equal to one with real…

Q: 2. Consider vectors of the form 2y + 5y 4y 3x + + 5z] 2х 8z ,x, y,z are real # - W = -x + 7z y +…

A: Solve the following

Q: #6. Let W be the subspace of R³ spanned by the two vectors (1, – 1,1) and (3, – 2,0). Which one of…

Q: Find the matrix A, relative to the standard basis {e1, e2, e3, e4}, of t

A: Given, W be the subspace of ℝ4 spanned by the vectors 1-11-1 and…

Q: What is the dimension of P₂, the vector space of all polynomials of degree at most two? b. Find the…

Q: Consider the vector space V over the real numbers, consisting of all polynomials of degree less than…

Q: 4. Let W be the subspace of R³ defined by the equation 2x – y +3z = 0. (a) I Find a basis {V1,V2} of…

Q: V is a 4-dimensional vector space with basis B = (v1, V2 , V3 , V4). T:V → V is linear…

Q: two operation (x, y) + (u, v) = (x+u, yv), for all (x, y), (u, v) € V a(x, y) = (ax, ya), for all…

Q: Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with the…

Q: Let W = {(x1, x2, x3) = R³ : x₁ + 2x2 − 3x3 = 0} x1 (a) Find a basis of W. (b) Give an example of a…

A: Answer follows

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

100%

Let V be the vector space P(t) of all polynomials in t over R, with inner product defined by
(f(t), g(t)) = So f(t)g(t) dt. Let W be the subspace P2(t) of V.
(a) Show that B
{1,2t – 1,6t2 – 6t + 1} is an orthogonal basis for W.
(b) Find the projection of f(t) = t³ onto W.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

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

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with the structure of an inner product space by setting (f,g):= f(t)g(t)dt. Produce an orthonormal basis for V by applying the Gramm-Schmidt orthogonalisation process to the basis (1, x, x²) of V.
If V is the subspace spanned by (1, 1, 0, 1) and (0, 0, 1, 0), find (a) a basis for the orthogonal complement V-L. (b) the projection matrix P onto V. (c) the vector in V closest to the vector b = (0, 1, 0, -1) in V¹.
Find a basis {p(r), q(x)} for the vector space {f(x) e P2[z] | f' (-8) = f(1)} where P2[T] is the vector space of polynomials in x with degree at most 2.
Consider the operator Son the vector space by S(a+bx) = - a+b+ ( a + 2b) x A) Pick a basis B = { b₁,b₂3. Find the minimal polynomials NT, b, (X), Nr, 0₂ (x), and Ns (x) R₁ [x] given B) Show that S is cyclic by finding a vector v such that
Let W be the subspace of R3 with orthonormal basis {w1, w2}, where
Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?
Let V₁ = ,V₂ = 1 and let W be the subspace of R¹ spanned by V₁ and v2. (a) Convert {V1, V2} into an orhonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = {}. (b) Find the projection of b = 1 onto W (c) Find two linearly independent vectors in R¹ perpendicular to W. Vectors = { Note: You can earn partial credit on this problem.
Let f(z) = -7, g(z) = -3z - 9 and h(z) = 8z". Consider the inner procuct (p, 9) in the vector space P(R) of polynomials. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace spanned by the functions f(2) 9(z), and h(z). {
TRUE or FALSE: If C = {u1,..., u,} is an orthonormal basis for a subspace H of R", and if y is a vector in H, then the coordinate vector of y with y ui y· u2 respect to the basis C is givén by [y]c = y up