Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
We have solutions for your book!
See our solution for Question 23E from Chapter 1.5 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. A homogeneous equation is always consistent. b. The equation Ax = 0 gives an explicit description of its solution set.
Step-by-Step Solution
Given Information
We are given with some statements that we have to prove whether they are True or False
Step-1: (a)
Statement: A homogeneous equation is always consistent.
The homogeneous system, where A is an matrix, and is the zero vector in has at least one solution, namely, As stated on Page-43, of the textbook, A system of linear equations is said to be homogeneous if it can be written in the form $Ax=0$, where A is an $m \times n$ matrix and 0 is the zero vector in $R^m$. Such a system $Ax = 0$ always has at least one solution, namely, $x=0$
The Statement is True
Step-2: (b)
Statement: The equation $Ax = 0$ gives an explicit description of its solution set.
The equation $Ax = 0$ gives an implicit description of its solution set
The Statement is False
Step-3: (c)
Statement: The homogeneous equation $Ax = 0$ has the trivial solution if and only if the equation has at least one free variable
As stated on page 44 of the textbook, The homogeneous equation $Ax = 0$ has a non-trivial solution if and only if the equation has at least one free variable
The Statement is False
Step-4: (d)
Statement: The equation ${\bf{x}} = {\bf{p}} + t{\bf{v}}$ describes a line through v parallel to p..
The equation ${\bf{x}} = {\bf{p}} + t{\bf{v}}$ is the equation of the line passing through $p$ and is parallel to $v$.
The Statement is False
Step-5: (e)
Statement: The solution set of $Ax = b$ is the set of all vectors of the form ${\bf{w}} = {\bf{p}} + {{\bf{v}}_{\bf{h}}}$, where $v_h$ is any solution of the equation $Ax = 0$.
The statement is true only when there exists some vector $p$ such that $A{\bf{p}} = {\bf{b}}$
The Statement is True
We are given with some statements that we have to prove whether they are True or False
Step-1: (a)
Statement: A homogeneous equation is always consistent.
The homogeneous system, where A is an matrix, and is the zero vector in has at least one solution, namely, As stated on Page-43, of the textbook, A system of linear equations is said to be homogeneous if it can be written in the form $Ax=0$, where A is an $m \times n$ matrix and 0 is the zero vector in $R^m$. Such a system $Ax = 0$ always has at least one solution, namely, $x=0$
The Statement is True
Step-2: (b)
Statement: The equation $Ax = 0$ gives an explicit description of its solution set.
The equation $Ax = 0$ gives an implicit description of its solution set
The Statement is False
Step-3: (c)
Statement: The homogeneous equation $Ax = 0$ has the trivial solution if and only if the equation has at least one free variable
As stated on page 44 of the textbook, The homogeneous equation $Ax = 0$ has a non-trivial solution if and only if the equation has at least one free variable
The Statement is False
Step-4: (d)
Statement: The equation ${\bf{x}} = {\bf{p}} + t{\bf{v}}$ describes a line through v parallel to p..
The equation ${\bf{x}} = {\bf{p}} + t{\bf{v}}$ is the equation of the line passing through $p$ and is parallel to $v$.
The Statement is False
Step-5: (e)
Statement: The solution set of $Ax = b$ is the set of all vectors of the form ${\bf{w}} = {\bf{p}} + {{\bf{v}}_{\bf{h}}}$, where $v_h$ is any solution of the equation $Ax = 0$.
The statement is true only when there exists some vector $p$ such that $A{\bf{p}} = {\bf{b}}$
The Statement is True