A question laws of the inverse

In summary, the conversation discusses the connection between the invertibility of a matrix and its dimension. Linear maps are mentioned as a way to understand this connection. The conversation also includes a link to a specific section of a webpage explaining the definition of a kernel and image, and how they relate to the rank-nullity theorem.
  • #1
transgalactic
1,395
0
in this question my book makes a certain presumption

that some how the ability for the matrix to be inversible
connects to its dim Im

can you say how it works in this question?

http://img441.imageshack.us/my.php?image=img86041mw3.jpg
 
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  • #2
Have you learned about linear maps yet? There are some nice relations such as these which will help you out here.
 
  • #3
it doesn't help me much in solving this question
can you please be more spesifiv about where to read in this page
because there are a lot of theory
which i already know

can you answer to my original question
 
  • #4
anyone??
 
  • #5
I linked to a very specific section of that page, namely the one stating
If [itex]f : V \to W[/itex] is linear, we define the kernel and the image or range of f by
:[tex]\operatorname{ker}(f)=\{\,x\in V:f(x)=0\,\}[/tex]
:[tex]\operatorname{im}(f)=\{\,w\in W:w=f(x),x\in V\,\}[/tex]
ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula, known as the rank-nullity theorem, is often useful:
[tex]\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ) [/tex]

Now if you have some knowledge of linear maps and such (which I presume you do, or should have at least, because the book uses terms like kernel and image in the first place), the statement will immediately follow.
 

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