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 17E from Chapter 2.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 17E
Chapter:
Problem:
In Exercises 17 and 18, mark each statement True or False. Justify each answer. Here A is an m × n matrix. a. is a basis for a subspace H and if are the coordinates of x relative to the basis b. Each line in is a one-dimensional subspace of .
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: (a)
As given by the definition in the textbook, the statement is True.
The statement is True
Step-2: (b)
one dimensional subspace of $R^n$ is Span(v) where v is a non zero vector in $R^n$. Span(v) is a line passing through the origin and parallel to v. Therefore, the line that does not pass through origin is not a subspace of R^n.
The statement is False
Step-3: (c)
The pivot columns of A are linearly independent and span the column space of A. So, pivot columns of A form a basis of Col A.
Hence the number of pivot columns of A is equal to dimension of Col A
The statement is True
Step-4: (d)
By the rank theorem, \[ \operatorname { rank } A + \operatorname { dim } \mathrm { Nul } A = n \] Rank of the matrix A is equal to dimension of the column space of A. Therefore
The statement is True
Step-5: (e)
By the basis theorem,
The statement is True
We are given with some statements that we have to find whether they are True or false
Step-1: (a)
As given by the definition in the textbook, the statement is True.
The statement is True
Step-2: (b)
one dimensional subspace of $R^n$ is Span(v) where v is a non zero vector in $R^n$. Span(v) is a line passing through the origin and parallel to v. Therefore, the line that does not pass through origin is not a subspace of R^n.
The statement is False
Step-3: (c)
The pivot columns of A are linearly independent and span the column space of A. So, pivot columns of A form a basis of Col A.
Hence the number of pivot columns of A is equal to dimension of Col A
The statement is True
Step-4: (d)
By the rank theorem, \[ \operatorname { rank } A + \operatorname { dim } \mathrm { Nul } A = n \] Rank of the matrix A is equal to dimension of the column space of A. Therefore
The statement is True
Step-5: (e)
By the basis theorem,
The statement is True