91Ó°ÊÓ

In order to confirm that a set is a vector space, we must demonstrate that it satisfies specific criteria, one of which is closure under addition.
This property means that if you take any two elements from the set and add them together, the result should also be in the set. Let's consider two elements, \((u_1, v_1)\) and \((u_2, v_2)\), from the Cartesian product of vector spaces \(U \times V\).
According to the rules of vector addition in \(U \times V\), you add these elements component-wise: \((u_1, v_1) + (u_2, v_2) = (u_1 + u_2, v_1 + v_2)\).

Since both\(u_1 + u_2\) belongs to \(U\) and \(v_1 + v_2\) belongs to \(V\), the sum \((u_1, v_1) + (u_2, v_2)\) definitely remains in \(U \times V\).
Thus, \(U \times V\) is closed under addition, meeting this criterion of being a vector space.

Vector spaces are defined by several axioms that must be honored for a set to be classified as such. Let’s briefly go over what these entail in the context of \(U \times V\).

• **Additive Identity:** There exists an additive identity, \((\mathbf{0}_U, \mathbf{0}_V)\), such that for any \((u, v)\) in \(U \times V\), we have \((u, v) + (\mathbf{0}_U, \mathbf{0}_V) = (u, v)\).
• **Additive Inverse:** Each element \((u, v)\) has an inverse \((-u, -v)\) such that \((u, v) + (-u, -v) = (\mathbf{0}_U, \mathbf{0}_V)\).
• **Associativity of Addition:** For any \((u_1, v_1)\), \((u_2, v_2)\), \((u_3, v_3)\), it holds that \(((u_1, v_1) + (u_2, v_2)) + (u_3, v_3) = (u_1, v_1) + ((u_2, v_2) + (u_3, v_3))\).
• **Distributive Properties:** For scalars \(a\) and \(b\), and vectors \((u, v)\), the operations \(a((u, v) + (u', v')) = a(\mathbf{u} + \mathbf{u'}) \) and \((a + b)(u, v) = a(u, v) + b(u, v)\) hold true.

By meeting these and the additional axioms, \(U \times V\) is verified to be a vector space.

The Cartesian product is a mathematical concept that combines two sets into ordered pairs. When dealing with vector spaces \(U\) and \(V\), the Cartesian product is denoted as \(U \times V\).
It consists of all possible pairs \((\mathbf{u}, \mathbf{v})\), where \(\mathbf{u}\) is in \(U\) and \(\mathbf{v}\) is in \(V\).

In practical terms, the Cartesian product forms a new vector space, where elements are not simply numbers but pairs of vectors from each set. This structure allows us to explore and visualize relationships between vector spaces and their interactions.
In our example, proving \(U \times V\) is a vector space involves showing that this set of pairs meets all required vector space axioms.
As each of the properties and operations in \(U\) and \(V\) hold on their own, they collectively validate \(U \times V\) as a vector space.

Scalar multiplication is a fundamental operation in linear algebra that involves multiplying a vector by a scalar (a constant from the field over which the vector space is defined).
In \(U \times V\), let a scalar be \(c\) and consider an arbitrary vector \((u, v)\).
Scalar multiplication is defined by \(c(u, v) = (cu, cv)\).
The result \((cu, cv)\) is another element of \(U \times V\) because of the closure properties of the original vector spaces \(U\) and \(V\).
Additional important properties related to scalar multiplication for vector spaces include:

• **Compatibility:** For any scalars \(a\) and \(b\), \(a(b(u, v)) = (ab)(u, v)\).
• **Distributive over Vectors and Scalars:** \(a((u, v) + (u', v')) = a(u, v) + a(u', v')\) and \((a + b)(u, v) = a(u, v) + b(u, v)\).

Through these properties, scalar multiplication not only demonstrates closure but also interacts reliably with addition within \(U \times V\), reinforcing the vector space structure.