Linear Algebra : Linear Mapping
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Example Question #1 : Linear Mapping
That last question dealt with isomorphism. This question is meant to point out the difference between isomorphism and homomorphisms.
A homomorphism is a mapping between vector spaces that
Preserves vector addition and is injective
onto and 1-to-1
Preserves vector addition and scalar multiplication
Preserves scalar multiplication and is onto
Preserves vector addition and scalar multiplication
By definition a homomorphism is a mapping that preserves vector addition and scalar multiplication.
Compare this to the previous problem. An isomorphism is a homomorphism that is also 1-to-1 and onto. Therefore isomorphism is just a special homomorphism. In other words, every isomorphism is a homomorphism, but not all homomorphisms are an isomorphisms.
Example Question #2 : Linear Mapping
Consider the mapping 
(Hint: Think about dimension's role in isomorphism)
No
not enough information
Yes
No
No, f, cannot be an isomorphism. This is because 
Example Question #3 : Linear Mapping
Isomorphism is an important concept in linear algebra. To be able to tell if a mapping is isomorphic, it is important to be able to know what an isomorphism is.
Let f be a mapping between vector spaces V and W. Then a mapping f is an isomorphism if it is
Preserves vector addition
Onto (surjective)
All answers.
Preserves scalar multiplcation
1-to-1 (injective)
All answers.
An isomorphism is homomorphism (preserves vector addition and scalar multiplcation) that is bijective (both onto and 1-to-1). Therefore an isomorphism is a mapping that is
1) onto
2) 1-to-1
3) Preserves vector addition
4) Preserves scalar multiplcation
Example Question #1 : Linear Mapping
In the previous question, we said an isomorphism cannot be between vector spaces of different dimension. But are all homomorphisms between vector spaces of the same dimension an isomorphism?
Consider the homomorphism 
Not enough information
No
Yes
Not enough information
The answer is not enough information. The reason is that it could be an isomorphism because it is between vector spaces of the same dimension, but that doesn't mean it is.
For example:
Consider the zero mapping f(x,y)= (0,0).
This mapping is not onto or 1-to-1 because all elements go to the zero vector. Therefore it is not an isomorphism even though it is a mapping between spaces with the same dimension.
Another example:
Consider the identity mapping f(x,y) = (x,y)
This is an isomorphism. It clearly preserves structure and is both onto and 1-to-1.
Thus f could be an isomorphism (example identity map) or it could NOT be an isomorphism ( Example the zero mapping)
Example Question #2 : Linear Mapping
Let f be a mapping such that
Let f be defined such that
Is f 1-to-1 and onto?
No, it is 1-to-1 and not onto
No, it is not 1-to-1 but it is onto
Yes
No, it is not 1-to-1 and not onto
No, it is not 1-to-1 and not onto
f is not 1-to-1 and it is not onto.
f is not onto because all of 
f is not 1-to-1. For example, the vector (1,1) and (1,0) both go to the same vector.
Ie f(1,1) = f(1,0). Therefore f is not 1-to-1.
Example Question #4 : Linear Mapping
Let f be a mapping such that
Let f be defined such that
Is f an isomorphism?
(Hint: Consider the zero vector)
No, it is not onto
No, it is not a homomorphism
Yes
No, it is not 1-to-1
No, it is not a homomorphism
f is 1-to-1 and onto but it is not a homomorphism. Therefore it is not an isomorphism. To see this consider f(0,0) = (0,5)
A homomorphism always takes the zero vector to the zero vector. This particular mapping does not. Thus it does not preserve structure ie not a homomorphism.
Example Question #5 : Linear Mapping
The last question showed us isomorphisms must be between vector spaces of the same dimension. This question now asks about homomorphisms.
Consider the mapping 
Yes
not enough information
No
Yes
The answer is yes. There is no restriction on dimension for homomorphism like there is for isomorphism. Therefore f could be a homomorphism, but it is not guaranteed.
Example Question #1 : Linear Mapping
Let f be a homomorphism from 
(Hint: look at the dimension of the domain and co-domain)
No
Not enough information
Yes
No
No, f can not be 1-to-1. The reason is because the domain has dimension 3 but the co-domain has dimension of 2. A mapping can not be 1-to-1 when the the dimension of the domain is greater than the dimension of the co-domain.
Example Question #2 : Linear Mapping
Often we can get information about a mapping by simply knowing the dimension of the domain and codomain.
Let f be a mapping from 
(Hint look at the dimension of the domain and codomain)
No
Yes
Not enough information
No
No, f cannot be onto. The reason is because the dimension of the domain (2) is less than the dimension of the codomain(3).
For a function to be onto, the dimension of the domain must be less than or equal to the dimension of the codomain.
Example Question #3 : Linear Mapping
The previous two problems showed how the dimension of the domain and codomain can be used to predict if it is possible for the mapping to be 1-to-1 or onto. Now we'll apply that knowledge to isomorphism.
Let f be a mapping such that 
Preserve scalar multiplication
t-to-1
Onto
Preserve vector addition
Onto
f cannot be onto. The reason is because the domain, V, has a dimension less than the dimension of the codomain, W.
f can be 1-to-1 since the dimension of V is less-than-or-equal to the dimension of W. However, just because f can be 1-to-1 based off its dimension does not mean it is guaranteed.
f preserves both vector addition and scalar multiplication because it was stated to be a homomorphism in the problem statemenet. The definition of a homomorphism is a mapping that preserves both vector addition and scalar multiplication.
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