91Ó°ÊÓ

To explore the notion of closure under addition in vector spaces, let's consider what it means in simpler terms. A vector space is considered closed under addition if, when you take any two vectors within that space, their sum is also a vector that falls within the same space. Using the exercise as an example, let's evaluate whether the given set of vectors, denoted as \( W \), meets this criterion.

When you add vectors \( \mathbf{u} = \begin{bmatrix} 3a_1 + b_1 \ 4 \ a_1 - 5b_1 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 3a_2 + b_2 \ 4 \ a_2 - 5b_2 \end{bmatrix} \) from set \( W \), their resultant sum is \[ \begin{bmatrix} 3(a_1 + a_2) + (b_1 + b_2) \ 8 \ (a_1 + a_2) - 5(b_1 + b_2) \end{bmatrix} \].

However, we notice that the second component is a fixed value, 8, instead of being the result of the addition of parameters like the other components. This disrupts the expected structural form and prevents \( W \) from being closed under addition, meaning it fails one of the core requirements of being a vector space.

Another fundamental property of vector spaces is closure under scalar multiplication. This means if you take any vector from the space and multiply it by any scalar (real number), the resulting vector should still belong to the same set. Let's see how \( W \) behaves with respect to this requirement.

Consider any vector \( \mathbf{u} = \begin{bmatrix} 3a + b \ 4 \ a - 5b \end{bmatrix} \) in \( W \) and a scalar \( k \). When you multiply these, the resultant vector is \[ \begin{bmatrix} k(3a + b) \ k imes 4 \ k(a - 5b) \end{bmatrix} = \begin{bmatrix} 3ka + kb \ 4k \ ka - 5kb \end{bmatrix} \].

The problem arises in the second component, which transforms into \( 4k \), still a constant form, merely scaled by \( k \). For a set to maintain closure under scalar multiplication, each component of the resulting vector should retain its original form, contingent upon the vectors \( a \), \( b \), and \( c \) being arbitrary. This inconsistency results in \( W \) failing the scalar multiplication closure criterion.

Vector spaces must adhere to a series of eight properties known as axioms. These axioms define the structural rules for how vector addition and scalar multiplication should behave consistently within the space. Here are the essentials:

• Closure under addition and scalar multiplication, as we've already discussed.
• Associativity and commutativity of vector addition.
• Existence of a zero vector (an additive identity).
• Existence of additive inverses (negatives) for each vector.
• Distributive properties over vector addition and scalar multiplication.
• Compatibility of scalar multiplication with field multiplication.
• Existence of a multiplicative identity (one).

For \( W \), the inability to satisfy closure under addition and scalar multiplication automatically means it cannot satisfy all the vector space axioms. If even one axiom is violated, the set \( W \) cannot be classified as a vector space. This highlights how crucial each axiom is to the overall structure and properties of vector spaces.