# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Tue, 21 May 2013 10:46:43 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Row Rank and Col Rank Unchanged by Row and Col OperationsPosted: Tue, 12 Oct 2010 06:07:52 UTC
 Member

Joined: Tue, 24 Mar 2009 04:48:47 UTC
Posts: 40
In my linear algebra text, there is a theorem that says row operations and column operations don't affect the dimension of the column space and row space of a matrix. If I were solving a system of linear equations, will collumn operations mess up the system. I am confused about what this theorem means?

_________________
Euler is God!

Top

 Post subject: Re: Row Rank and Col Rank Unchanged by Row and Col OperationPosted: Tue, 12 Oct 2010 13:05:14 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
srjcstud wrote:
In my linear algebra text, there is a theorem that says row operations and column operations don't affect the dimension of the column space and row space of a matrix. If I were solving a system of linear equations, will collumn operations mess up the system. I am confused about what this theorem means?

No, this isn't how you are supposed to think about it.

If you sole purpose is to solve a linear system in the form , then column operations are basically useless, because you are not interested in solving for values of e.g. , but value of .

Similarly, if your sole purpose is to solve a linear system of the form , then row operations are basically useless for a similar reason.

Instead, this is a theorem because it asserts two different ways of defining rank (i.e. dimension of row space, dimension of column space) are actually the same thing, so we can speak of the rank of a matrix without specifying whether we mean the dimension of row space, or the dimension of column space. Fom the proof of this we get, for example, the rank-nullity formula.

_________________

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous