Let Ŝ be the subspace of C[a, b] spanned by {e², xe², x²e²}. (a) Show that B = {e², xe², x²e²} is a basis of S. (b) Let D: S –→ S be the differentiation operator. Find (D]Bb, the matrix of D with respect to the ordered basis B. Find all the eigenvectors of [D]bb and hence find all the solutions in S of the differential equation df f. dr

Let Ŝ be the subspace of C[a, b] spanned by {e², xe², x²e²}. (a) Show that B = {e², xe², x²e²} is a basis of S. (b) Let D: S –→ S be the differentiation operator. Find (D]Bb, the matrix of D with respect to the ordered basis B. Find all the eigenvectors of [D]bb and hence find all the solutions in S of the differential equation df f. dr

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 S be the subspace of C[a, b] spanned by {e", xe",x²e²}.
(a) Show that B =
{e*, xe", x²e"} is a basis of S.
(b) Let D: S → S be the differentiation operator. Find [D]bB, the matrix of
D with respect to the ordered basis B. Find all the eigenvectors of [D]bB and
hence find all the solutions in S of the differential equation
df
f.
(c) Given that C = {(1 – x)e", 2e*, (1 + x²)e*} is also an ordered basis of S.
Find the change of basis matrix from B to C and describe how one obtains
from this the matrix [D]cc.

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