# Linear Algebra : Linear Mapping

## Example Questions

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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 scalar multiplication

onto and 1-to-1

Preserves scalar multiplication and is onto

Preserves vector addition and is injective

Preserves vector addition and scalar multiplication

Explanation:

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 . Can f be an isomorphism?

(Hint: Think about dimension's role in isomorphism)

not enough information

Yes

No

No

Explanation:

No, f, cannot be an isomorphism. This is because  and  have different dimension. Isomorphisms cannot exist between vector spaces of different dimension.

### 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

1-to-1 (injective)

Onto (surjective)

Preserves scalar multiplcation

Explanation:

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

4) Preserves scalar multiplcation

### Example Question #4 : 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 . Is f an isomorphism?

No

Not enough information

Yes

Not enough information

Explanation:

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 #5 : Linear Mapping

Let f be a mapping such that

Let f be defined such that

Is f 1-to-1 and onto?

Yes

No, it is not 1-to-1 and not onto

No, it is 1-to-1 and not onto

No, it is not 1-to-1 but it is onto

No, it is not 1-to-1 and not onto

Explanation:

f is not 1-to-1 and it is not onto.

f is not onto because all of is not in the image of f. For example, the vector (1,1) is not in the image of f.

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 #6 : Linear Mapping

Let f be a mapping such that

Let f be defined such that

Is f an isomorphism?

(Hint: Consider the zero vector)

Yes

No, it is not 1-to-1

No, it is not a homomorphism

No, it is not onto

No, it is not a homomorphism

Explanation:

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 #7 : 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 . Can f be a homomorphism?

No

Yes

not enough information

Yes

Explanation:

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 #8 : Linear Mapping

Let f be a homomorphism from  to . Can f be 1-to-1?

(Hint: look at the dimension of the domain and co-domain)

Yes

Not enough information

No

No

Explanation:

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 #9 : 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   to . Can f be onto?

(Hint look at the dimension of the domain and codomain)

No

Not enough information

Yes

No

Explanation:

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 #10 : 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 . Also the vector space V has dimension 4 and the vector space W has dimension 8. What property of isomorphism can f NOT satisify.

t-to-1

Preserve scalar multiplication

Onto

Onto

Explanation:

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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