Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 23E from Chapter 4.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 23E
Chapter:
Problem:
In Exercises 23 and 24, mark each statement True or False. Justify each answer. a. If f is a function in the vector space V of all real-valued functions on and if
Step-by-Step Solution
Given Information
We are given with some statements, that we have to find whether they are true or False
Step-1: Properties of Subspace
(a) The zero vector is in H
(b) The each u and v in H, the sum of u+v is in H
(c) For each u in H and a scalar c, the vector $cu$ is in H
Step-2: (a)
Even though $f(t)$ is zero for some $t$, it is not necessary that $f(t)$ be zero for all $t$. Hence
The statement is False
Step-3: (b)
The definition of vector says that it is a directed arrow with a specific magnitude in the three-dimensional space. The space may or may not be a three-dimensional space. Hence
The statement is False
Step-4: (c)
As given in step-1, the subspace should satisfy all three conditions including the zero vector condition. Hence
The statement is False
Step-5: (d)
A subset of a vector space is a subspace if it is itself a vector space
The statement is True
Step-5: (e)
Discrete signals are used for space shuttles
The statement is False
We are given with some statements, that we have to find whether they are true or False
Step-1: Properties of Subspace
(a) The zero vector is in H
(b) The each u and v in H, the sum of u+v is in H
(c) For each u in H and a scalar c, the vector $cu$ is in H
Step-2: (a)
Even though $f(t)$ is zero for some $t$, it is not necessary that $f(t)$ be zero for all $t$. Hence
The statement is False
Step-3: (b)
The definition of vector says that it is a directed arrow with a specific magnitude in the three-dimensional space. The space may or may not be a three-dimensional space. Hence
The statement is False
Step-4: (c)
As given in step-1, the subspace should satisfy all three conditions including the zero vector condition. Hence
The statement is False
Step-5: (d)
A subset of a vector space is a subspace if it is itself a vector space
The statement is True
Step-5: (e)
Discrete signals are used for space shuttles
The statement is False