# A question laws of the inverse

• transgalactic
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.
transgalactic
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

Have you learned about linear maps yet? There are some nice relations such as these which will help you out here.

it doesn't help me much in solving this question
because there are a lot of theory

can you answer to my original question

anyone??

I linked to a very specific section of that page, namely the one stating
If $f : V \to W$ is linear, we define the kernel and the image or range of f by
:$$\operatorname{ker}(f)=\{\,x\in V:f(x)=0\,\}$$
:$$\operatorname{im}(f)=\{\,w\in W:w=f(x),x\in V\,\}$$
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:
$$\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V )$$

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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