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May 19th, 2009, 07:19 AM  #1 
Newbie Joined: May 2009 Posts: 20 Thanks: 0  linear algebra vector spaces
i really need help with this question: Let V be a finitedimensional vector space. suppose that T : V>V is a linear operator. Show that T is injective if and only if T is surjective. and are there any results for sets and functions? 
May 20th, 2009, 07:53 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: linear algebra vector spaces
Use the ranknullity theorem  i.e. the one that says You can find by noting that if is injective, then and using the properties of a linear map. 
May 20th, 2009, 09:23 AM  #3 
Newbie Joined: May 2009 Posts: 20 Thanks: 0  Re: linear algebra vector spaces
sorri could u possibly expand you reply im not getting you cheers 
May 20th, 2009, 09:43 AM  #4 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: linear algebra vector spaces
The ranknullity theorem states that for a linear map the dimension of the kernel of and the dimension of the image of sum to the dimension of The kernel of is defined to be the space of vectors such that The image of is the space of vectors such that there exists an such that Clearly, a linear operator is surjective iff You need to show that for we have iff is injective, using the ranknullity theorem. To find the kernel of an injective map, note that if is injective, then for all or If you let you therefore have that is injective iff for all Then use the properties of a linear operator to show that Once this is established, you get the desired result by a simple application of the theorem. 
May 26th, 2009, 01:48 AM  #5 
Newbie Joined: May 2009 Posts: 20 Thanks: 0  Re: linear algebra vector spaces
thankyou so much for your help it was much appreciated 

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algebra, linear, spaces, vector 
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