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**Vector Spaces**

**Definition:** A *vector space* is a set V on which two operations + and **Â·** are defined, called *vector addition* and *scalar multiplication.*

The operation + (vector addition) must satisfy the following conditions:

*Closure:* If **u** and **v** are any vectors in V, then the sum **u** + **v** belongs to V.**(1)** *Commutative law:* For all vectors **u** and **v** in V, **u** + **v** = **v** + **u****(2)** *Associative law:* For all vectors **u**, **v**, **w** in V, **u** + (**v** + **w**) = (**u** + **v**) + **w****(3)** *Additive identity:* The set V contains an *additive identity* element, denoted by **0**, such that for any vector **v** in V, **0** + **v** = **v** and **v** + **0** = **v**.**(4)** *Additive inverses:* For each vector **v** in V, the equations **v** + **x** = **0** and **x** + **v** = **0** have a solution **x** in V, called an *additive inverse* of **v**, and denoted by - **v**.

The operation **Â·** (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:*Closure:* If **v** in any vector in V, and c is any real number, then the product c **Â·** **v** belongs to V.**(5)** *Distributive law:* For all real numbers c and all vectors **u**, **v** in V, c **Â·** (**u** + **v**) = c **Â·** **u** + c **Â·** **v****(6)** *Distributive law*: For all real numbers c, d and all vectors **v** in V, (c+d) **Â·** **v** = c **Â·** **v** + d **Â·** **v****(7)** *Associative law*: For all real numbers c,d and all vectors **v** in V, c **Â·** (d **Â·** **v**) = (cd) **Â·** **v****(8)** *Unitary law*: For all vectors **v** in V, 1 **Â·** **v** = **v**

**Subspaces**

**Definition:** Let V be a vector space, and let W be a subset of V. If W is a vector space with respect to the operations in V, then W is called a *subspace* of V.

**Theorem:** Let V be a vector space, with operations + and **Â·**, and let W be a subset of V. Then W is a subspace of V if and only if the following conditions hold.**Sub0** *W is nonempty*: The zero vector belongs to W.**Sub1** *Closure under +*: If **u** and **v** are any vectors in W, then **u + v** is in W.**Sub2** *Closure under Â·*: If

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