# Freshman Linear Algebra

## Vector Spaces and Subspaces

Definition: A vector space is a nonempty set of objects, called vectors, on which are defined two operations, called additon and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. The axioms must holr for all vectors u, v, and w in V and for all scalars c and d.

1. The sum of u and v, denoted u+v, is in V.
2. u+v=v+u
3. (u+v)+w=u+(v+w)
4. There is a zero vector 0 in V such that u+0=u *0 is unique.
5. For each u in V, there is a vector -u in V such that u+(-u)=0. -u is unique and is called the negative of u.
6. The scalar multiple of u by c, denoted cu, is in V.
7. c(u+v)=cu+cv.
8. (c+d)u=cu+du.
9. c(du)=(cd)u.
10. 1⋅u=u

*for each u in V and scalar c:

• 0u=0
• c0=0
• -u=(-1)u

Example 1: The spaces R², where n≥1, are the premier examples of vector spaces. The geometric intuiton developed for R³ will help you understand and visualize many concepts throught the chapter.

Example 2: Let V be the set of all arrows(directed line segments) in three dimensional space, with two arrows regarded as equal if they have the same length and point in the same direction. Define additon by parallelogram rule and for each v in V, define cv to be the same direction. Define additon by

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