# Proof involving vector spaces and linear transformations

• killpoppop
In summary: Hm, so you want to prove that T(v)+T(-v)=0. And what you're doing is taking each term and adding it together, so in this case it would be T(v)+T(-v)+T(-v)+T(-v)=0.
killpoppop
1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v)

2. -T(v) denotes the additive inverse of T(v)

3. I think I'm really overcomplicating it =/ But i have

0v = T( v - v ) = T(v) + T(-v)

0v + -T(v) = T(v) + T(-v)

(T(v) + T(-v)) -T(v) = T(v) + T(-v)

then

T(-v) = T(v) + T(-v)

then i suppose it could go to

T(-v) = 0v

but that doesn't help I'm going round in circles. Basically i need a starting point.

Looks more like you need someone to tell you when to stop.

When you obtain 0=T(v)+T(-v), you're done. If T(-v) has that property, it is the additive inverse of T(v). You should however explain more clearly why T(v+(-v))=0.

Hm, are you allowed to use that 0x=0 for all x without proof? If not, it's easy to prove using the trick 0x=(0+0)x.

Last edited:
Cheers!

Right so when i get:

0v = T(v) + T(-v)

i just move the T(v) over because 0v=0?

giving

-T(v) = T(-v)

seems a bit too easy! I've already proved 0x=0 .

'You should however explain more clearly why T(v+(-v))=0.'

You shouldn't need to move anything over, you know that 0v = 0, therefore at the step where you have T(v) + T(-v) = 0 that means T(-v) is the additive inverse of T(v), remember what definition of additive inverse is!

killpoppop said:

'You should however explain more clearly why T(v+(-v))=0.'
I'm not sure if I can tell you anything new without giving you the complete solution. What you want to prove is that T(v)+T(-v)=0. And the proof of that goes like this:

T(v)+T(-v)=T(v+(-v))=...=0

Can you fill in the missing steps?

## 1. What is a vector space?

A vector space is a set of objects, called vectors, that can be added together and multiplied by scalars (numbers). The addition and multiplication operations must follow specific rules, such as closure, associativity, commutativity, and the existence of an identity element. A vector space is an important mathematical concept used in various fields, including physics, engineering, and computer science.

## 2. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the vector space structure. In other words, it is a function that satisfies the properties of linearity, such as preserving addition and scalar multiplication. Linear transformations are essential in understanding and solving problems involving vector spaces, such as finding solutions to systems of linear equations.

## 3. How can I prove that a given set is a vector space?

To prove that a set is a vector space, you need to show that it satisfies the ten properties of vector spaces. These properties include closure under addition and scalar multiplication, associativity, commutativity, the existence of an identity element, and the existence of inverse elements. You can also use the properties of vector spaces to show that a set is not a vector space by finding a counterexample.

## 4. How can I prove that a given function is a linear transformation?

To prove that a function is a linear transformation, you need to show that it satisfies the properties of linearity. These properties include preserving addition and scalar multiplication, as well as mapping the zero vector to the zero vector. You can also use the definition of a linear transformation to prove that a function is not linear by finding a counterexample.

## 5. What is the difference between a vector space and a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. In other words, a subspace is a smaller vector space contained within a larger vector space. It is important to note that a subspace must still satisfy all ten properties of vector spaces, including closure under addition and scalar multiplication. If a subset does not satisfy all the properties, it is not a subspace.

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