Is T(inv)(0) a Subspace of V and T(V) a Subspace of W?

In summary, the conversation is about defining linear maps T: V -> W and two subspaces, T(inv) (0) and T(V), in the vector spaces V and W respectively. The aim is to show that T(inv) (0) is a subspace of V and T(V) is a subspace of W. The conversation involves discussing the three axioms for a subspace and the importance of verifying them in order to determine if something is a subspace or not. The person asking for help is looking for a detailed response to ensure they have the correct answer.
  • #1
mang733
5
0
Could someone help me here please?

Let V and W be two vector spaces and T: V -> W be a linear map. Define

T(inv) (0) = { u element of V l T(u) = 0 }

where 0 is the zero element in W. Also define,

T(V) = { T(u) l u element of V},

the image of V under T. Show that T(inv) (0) is a subspace of V and T(V) is a subspace of W.

Your help is much appreciated!
 
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  • #2
Just chekc the axioms for a subspave are satisfied, and post your working.
 
  • #3
you mean the three axioms?
 
  • #4
since I want to make sure I have the correct answer, if someone has the time I appreciate a detailed response.
 
  • #5
Doing someone else's homework isn't very interesting to most people. So post what you've managed to verify. Something is a subspace if and only iof it satisfies the rules for being a subspace; how far have you been able to verify that the rules hold?
 

FAQ: Is T(inv)(0) a Subspace of V and T(V) a Subspace of W?

What is a linear map subspace?

A linear map subspace is a subset of a vector space that is closed under addition and scalar multiplication. In other words, if you take any two vectors in the subspace and add them together, the result will also be in the subspace. Similarly, if you multiply any vector in the subspace by a scalar, the result will also be in the subspace.

How do you prove that a subset is a linear map subspace?

To prove that a subset is a linear map subspace, you need to show that it satisfies the two properties of closure under addition and scalar multiplication. This can be done by taking two arbitrary vectors in the subset and showing that their sum and scalar multiples are also in the subset.

What is the significance of proving a subset to be a linear map subspace?

Proving a subset to be a linear map subspace is important because it allows us to understand the structure of a vector space and its subspaces. It also helps us to determine the dimension of the subspace and its relationship to the original vector space.

Can a subset be a linear map subspace of more than one vector space?

Yes, a subset can be a linear map subspace of more than one vector space. This is because the properties of closure under addition and scalar multiplication are independent of the vector space that the subset belongs to. As long as the subset satisfies these properties, it can be considered a linear map subspace of any vector space.

Are there any common mistakes to avoid when proving a subset to be a linear map subspace?

One common mistake to avoid is assuming that a subset is a linear map subspace without actually proving it. It is important to provide a clear and logical proof for the subset to be considered a linear map subspace. Another mistake is not checking for all possible vectors in the subset when showing closure under addition and scalar multiplication.

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