Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Linear and Abstract Algebra
What is the Corollary of the Nucleus and Image Theorem?
Reply to thread
Message
[QUOTE="pasmith, post: 6457671, member: 415692"] The problem is not to find an [itex]n[/itex] for which each case is possible, but to show that each case is possible for every [itex]n \geq 2[/itex] (With [itex]n = 1[/itex] the first is impossible, since there is at most one linearly independent vector in [itex]\mathbb{R}^1[/itex].) There are clearly two possibilities for [itex]n \geq 2[/itex]: Either [itex]F(u)[/itex] and [itex]F(v)[/itex] are linearly independent or they are not. If they are, we have case (1). So suppose they are not linearly independent. Then there exist non-zero scalars [itex]A[/itex] and [itex]B[/itex] such that [tex]0 = AF(u) + BF(v)[/tex]. What can you say about [itex]Au + Bv[/itex] in this case? The posbbility that [itex]F(u) = F(v) = 0[/itex] is not excluded. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Linear and Abstract Algebra
What is the Corollary of the Nucleus and Image Theorem?
Back
Top