Let P3 be the set of all polynomials of degree at most 3. That isP3 = {ao + ait + azt? + azt³ | ao,a1,a2, a3 E R}.With the zero polynomial po(t) = 0, the usual addition, and the usual scalar multiplication, P3 is a vector space.Let H = {bo + bit + b3t | bo, b1, b3 E R} be a subset of P3.(a) Give a nonzero polynomial h1 (t) in H, and a polynomial h2(t) in P3 which is not in H.(b) Show that H is a subspace of P3.(c) Give a basis for H (you must show why is it a basis for H). What is the dim(H)?

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Let P3 be the set of all polynomials of degree at most 3. That is P3 = {ao + at+ azt? + azt | ao, a1, 02, a3 E R}. With the zero polynomial po(t) = 0, the usual addition, and the usual scalar multiplication, P3 is a vector space. Let H = {bo + bit + b3t3 | bo, b1, b3 E R} be a subset of P3. (a) Give a nonzero polynomial h1(t) in H, and a polynomial h2(t) in P3 which is not in H. (b) Show that H is a subspace of P3. (c) Give a basis for H (you must show why is it a basis for H). What is the dim(H)?

Let P3 be the set of all polynomials of degree at most 3. That is P3 = {ao + at+ azt? + azt | ao, a1, 02, a3 E R}. With the zero polynomial po(t) = 0, the usual addition, and the usual scalar multiplication, P3 is a vector space. Let H = {bo + bit + b3t3 | bo, b1, b3 E R} be a subset of P3. (a) Give a nonzero polynomial h1(t) in H, and a polynomial h2(t) in P3 which is not in H. (b) Show that H is a subspace of P3. (c) Give a basis for H (you must show why is it a basis for H). What is the dim(H)?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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

Let P3 be the set of all polynomials of degree at most 3. That is
P3 = {ao + ait + azt? + azt³ | ao,a1,a2, a3 E R}.
With the zero polynomial po(t) = 0, the usual addition, and the usual scalar multiplication, P3 is a vector space.
Let H = {bo + bit + b3t | bo, b1, b3 E R} be a subset of P3.
(a) Give a nonzero polynomial h1 (t) in H, and a polynomial h2(t) in P3 which is not in H.
(b) Show that H is a subspace of P3.
(c) Give a basis for H (you must show why is it a basis for H). What is the dim(H)?

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