Answered: Complete each of the following…

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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Related questions

Question

Complete each of the following sentences by matching each of the letter options (e.g. (A), (B), ...) to a correct corresponding number option (e.g. (1), (2), ...).

A) For the matrix LaTeX: M:=\begin{pmatrix}0& -1\\ 1& 0 \end{pmatrix} we have LaTeX: \text{span}(\{I, M, M^{2}, M^{3},....,M^{n},...\}) is equal to

(B) Let LaTeX: x,\:x^2 be polynomial functions in LaTeX: \mathcal P_{2}(\mathbb F_2), the vector space of all polynomial functions of degree two or less with coefficients in LaTeX: \mathbb F_2. Then LaTeX: \mathrm{span}(x) is equal to

(D) Let LaTeX: T:V\to W be a linear transformation and LaTeX: S be a subset of LaTeX: V. Then LaTeX: T(\text{span}(S)) is equal to

(1) A strict subspace of LaTeX: \mathrm{span}(x,x^{2}) (that is a subspace of LaTeX: \mathrm{span}(x,x^{2}) which is not equal to LaTeX: \mathrm{span}(x,x^{2})).

(2) The space of LaTeX: 2\times 2 matrices with real entries and zero diagonal.

(3) LaTeX: M_{2,2}(\mathbb R)

(4) LaTeX: \mathrm{span}(x,x^{2})

(5) LaTeX: \mathrm{span}(1,x,x^2,x^3, \cdots)

(6) None of the options completes this phrase into a true statement.

(7) LaTeX: \text{span}(T(S))

(8) LaTeX: W

Complete each of the following sentences by matching each of the letter options (e.g. (A), (B), ...) to a
correct corresponding number option (e.g. (1), (2), ...).
-1
we have span({I,M, M² , M³,....,
M",...}) is
(A) For the matrix M::
1
equal to
(B) Let x, x2 be polynomial functions in P2 (F2), the vector space of all polynomial functions of
degree two or less with coefficients in F2. Then span(x) is equal to
(C) The vector space of all functions over real numbers is a subspace of
(D) Let T : V
→ W be a linear transformation and S be a subset of V. Then T(span(S)) is equal
to
(1) A strict subspace of span(x, x?) (that is a subspace of span(x, x2) which is not equal to
span(x, x²).
6.
(2) The space of 2 x 2 matrices with real entries and zero diagonal.
(3) M2,2 (IR)
(4) span(x, x²)
(5) span(1, x, x² , x³, … )
(6) None of the options completes this phrase into a true statement.

(6) None of the options completes this phrase into a true statement.
(7) span(T(S))
(8) W
A
2
В
[ Choose ]
C
[ Choose ]
[ Choose ]
>
>
>

Expert Solution

Introduction

The given questions are related with linear algebra.

We have to complete each of the following sentences by matching each of the letter options (e.g. (A),(B), ...)to a corresponding number option (e.g. (1) , (2), ...).

(A) For the matrix $M = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$ we have $s p a n (\{I, M, M^{2}, \dots, M^{n}, \dots\})$ is equal to

(B) Let $x, x^{2}$ be polynomial functions in $P_{2} (F_{2})$ , the vector space of all polynomial functions of degree two or less with coefficients in $F_{2}$ . Then $s p a n (x)$ is equal to