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You probably just need to pay more attention to the definitions, be more careful with your statements, and do more exercises. Getting a copy of the textbook would probably be a good start. 
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SUMMARY
The discussion centers on verifying whether the set W, defined as {f ∈ F(ℝ) | f(-x) = f(x) for all x ∈ ℝ}, is a subspace of the vector space F(ℝ). Participants confirm that the zero function is in W and demonstrate closure under addition using the properties of even functions. The confusion arises around closure under scalar multiplication, which is clarified by stating that for any scalar a, (af)(x) must also satisfy the even function condition. The importance of correctly identifying the zero vector and understanding the definitions of vector spaces is emphasized.
PREREQUISITES- Understanding of vector spaces and their properties
- Familiarity with function notation and operations in F(ℝ)
- Knowledge of even functions and their characteristics
- Basic principles of linear algebra, including subspace tests
- Study the definition of vector spaces, particularly F(ℝ)
- Learn about the properties of even functions and their implications in linear algebra
- Review scalar multiplication in the context of function spaces
- Practice problems involving subspace tests for various function sets
Students of linear algebra, particularly those struggling with concepts of vector spaces and function properties, as well as educators seeking to clarify these topics in their teaching.
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