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Vector addition is a fundamental concept in linear algebra which involves combining two or more vectors to form a new vector, known as the resultant vector. When adding two vectors, like vector v and vector w, you place the tail of w at the head of v, and then draw a vector from the tail of v to the head of w. This new vector represents the sum v + w.

It's important to understand that vector addition is commutative; that is, v + w is the same as w + v. Additionally, it follows the associative property, meaning that no matter how you group the vectors you’re adding, the resultant sum remains the same. For example, (v + u) + w is the same as v + (u + w), where u, v, and w are any vectors in the same space.

Vector subtraction can be viewed as a specific case of vector addition. To subtract a vector, like vector w, from another vector, like vector v, you simply add the opposite of vector w to v. The 'opposite' of a vector is the same vector with its direction reversed, also called its negation. Mathematically, this is represented as v - w being equivalent to v + (-w).

This operation shifts the position of the tail of w to the head of v, and creates a new vector that points from the head of w to the head of v. Subtraction of vectors is used to determine the distance between points in the space when they are represented as position vectors.

The associative property is a rule that applies to both addition and multiplication operations in various mathematical systems, including the realm of linear algebra. It states that the way in which operators are grouped in an expression does not affect the result of the operation. In the context of vector addition, this property ensures that when you add multiple vectors together, the grouping of these vectors does not matter.

Mathematically, if you have three vectors a, b, and c, the associative property of vector addition can be expressed as (a + b) + c = a + (b + c). This property is crucial when simplifying vector expressions and solving for vector identities, aiding in the understanding of complex vector operations. It simplifies computations and is a fundamental principle that underlies many proofs in linear algebra.

Proving theorems and identities in linear algebra often involves a series of logical steps that adhere to defined properties and operations within the field. Such proofs validate the truth of statements and are a cornerstone for understanding and working with vectors and matrices. For vector identities specifically, proofs typically involve demonstrating that the left-hand side (LHS) of an equation is equivalent to its right-hand side (RHS) under certain operations.

Using the properties of vector addition, subtraction, and associative property, linear algebra proofs become more manageable. They begin with known identities and, through a series of substitutions and simplifications, arrive at the desired result. These proofs not only validate the identity in question but also reinforce the interconnectedness of dot products, cross products, and other vector operations in linear algebra.