# Linear Maps

• Oct 15th 2009, 09:56 AM
GreenDay14
Linear Maps
I have a question here that reads:

Let T: V -> F be a linear map. Prove that if u belongs to V is not a null(T), then
V = nullT direct sum of {au |a belongs to F}

Could anyone give me some idea on where I should start? Any help would be greatly appreciated. Thanks.
• Oct 15th 2009, 10:15 AM
tonio
Quote:

Originally Posted by GreenDay14
I have a question here that reads:

Let T: V -> F be a linear map. Prove that if u belongs to V is not a null(T), then
V = nullT direct sum of {au |a belongs to F}

Could anyone give me some idea on where I should start? Any help would be greatly appreciated. Thanks.

You shoould explain stuff better so that people won't to guess what you meant: here V is a vector spcae over field F and T V --> F is a linear trasf., or much better and more widely knows as linear functional. Then...etc.

Well, this is simple: if dim_F (V) = n and since dim_F (F) = 1 , use now the well-known dimensions theorem for linear transformations between finite dimensional vector spaces.

Tonio
• Oct 15th 2009, 10:20 AM
GreenDay14
I apologize if I may have come off as a bit confusing, but I explained it to the best of my ability. With that said, would it be possible to elaborate a bit? The dimensions theorem is where I am having most of my trouble.
• Oct 15th 2009, 10:22 AM
tonio
Quote:

Originally Posted by GreenDay14
I apologize if I may have come off as a bit confusing, but I explained it to the best of my ability. With that said, would it be possible to elaborate a bit? The dimensions theorem is where I am having most of my trouble.

Fine, but then it didn't take you long to write back. Check now in your notes/books the dimensions theorem and how can you apply it to the present situation, then think and work a little and make some effort, and THEN write back.

Tonio
• Oct 15th 2009, 10:27 AM
GreenDay14
Actually, I thought the dim_F (V) = n and since dim_F (F) = 1 was also pretty standard, meaning I had already assumed this before posting, I have made attempts at this problem prior to posting as I am well aware that this is a Math HELP Forum. Please don't assume I did not try to work out this question.

I am still in complete disarray as to how to tackle this problem. Could someone please offer me some guidance. Thanks.