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 32E from Chapter 1.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 32E
Chapter:
Problem:
Could a set of three vectors in R4 span all of ℝ4? Explain. What about n vectors in ℝm when n is less than m?
Step-by-Step Solution
Given Information
We have to find whether the set of three vectors (in R^4) can span all of $R^4$. Moreover, we have to find about $n$ vectors in $R^m$.
Step-1:
The set of three vectors (in R^4) means a matrix of size $4 \times 3$. The matrix can have maximum of 3 pivot elements. Hence one of the row will remain devoid of any pivot element. Therefore, the columns of A can not span R^4.
Step-2:
As a general rule we can follow that, if there is a set of n vectors in $R^m$, it represents a matrix of $m$ rows and $n$ columns.
Therefore, the matrix can have only n pivot elements (less than m). So all the rows (m) will not have pivot element. So
The the set of n vectors can not span all of $R^m$, if $n < m$
We have to find whether the set of three vectors (in R^4) can span all of $R^4$. Moreover, we have to find about $n$ vectors in $R^m$.
Step-1:
The set of three vectors (in R^4) means a matrix of size $4 \times 3$. The matrix can have maximum of 3 pivot elements. Hence one of the row will remain devoid of any pivot element. Therefore, the columns of A can not span R^4.
Step-2:
As a general rule we can follow that, if there is a set of n vectors in $R^m$, it represents a matrix of $m$ rows and $n$ columns.
Therefore, the matrix can have only n pivot elements (less than m). So all the rows (m) will not have pivot element. So
The the set of n vectors can not span all of $R^m$, if $n < m$