Modern Algebra 3 Quizz

1. What is a homomorphism of vector spaces?

• A bijective linear transformation between two vector spaces.
• A linear transformation between two vector spaces that preserves the structure of the vector space.
• A non-linear transformation between two vector spaces.
• A transformation between two vector spaces that is only defined on a subset of the original vector space.

2. What is the definition of a homomorphism of vector spaces?

• A linear transformation between two vector spaces that takes the zero vector of one space to the zero vector of the other space.
• A linear transformation between two vector spaces that takes every vector of one space to a scalar multiple of itself in the other space.
• A linear transformation between two vector spaces that takes every vector of one space to a linearly independent vector in the other space.
• A non-linear transformation between two vector spaces.

3. Which of the following is a property of a homomorphism of vector spaces?

• It is always surjective.
• It is always injective.
• It is always bijective.
• It can be injective, surjective, or neither.

4. Can a homomorphism of vector spaces increase the dimension of the vector space?

• No
• Yes
• It depends on the basis chosen for the vector spaces.
• It depends on the underlying field.

5. Which of the following is a necessary condition for a linear transformation to be a homomorphism of vector spaces?

• It must be injective.
• It must be surjective.
• It must be bijective.
• It must preserve the structure of the vector space.

6. How can you determine if a linear transformation is a homomorphism of vector spaces?

• By checking if it preserves the structure of the vector space.
• By checking if it is injective.
• By checking if it is surjective.
• By checking if it is bijective.

7. What is the meaning of the term "kernel" in the context of homomorphisms of vector spaces?

• The subspace of the original vector space that is mapped to the zero vector of the target vector space.
• The subspace of the target vector space that is the image of the original vector space.
• The set of scalars that are mapped to the zero vector of the target vector space.
• The set of vectors in the original vector space that are mapped to the zero vector of the target vector space.