Linear Algebra (3)

A matrix with only one column is called a vector.

         1
V =  2
         3

V can also be expressed as V = [ 1 , 2 , 3 ]. Essentially, a vector is an ordered set of numbers. The set of all vectors with three entries is denoted by  ℝ^3. Given two vectors u and v in ℝ^3, their sum u + v is a vector w obtained by adding the corresponding entries of u and v. For example, if u = [ 1 , 2 , 3 ] and v = [ 2 , 3 , 4 ], then w = u + v = [ 1 , 2 , 3 ] + [ 2 , 3 , 4 ] = [ 1 + 2 , 2 + 3 , 3 + 4 ] = [ 3 , 5 , 7 ].

Given a vector u and a real number c, the scalar multiple of u by c is the vector c(u) obtained my multiplying each entry in u by c. The number c in c(u) is called a scalar.

Geometric Descriptions of ℝ^2
The following is a graph of the set of all vectors in ℤ^2 which is a subset of ℝ^2 such that each vector is not a scalar multiple of any other vector in the set and their components are between -8 to 8. The vectors that are in the set are marked with a purple dot. 

Vector addition geometrically amounts to the construction of a parallelogram.

Here are some algebraic properties of ℝ^n:

For all u, v, and w in ℝ^n and scalars c and d

1.) u + v = v + u

2.) (u + v) + w = u + (v + w)

3.) u + 0 = u

4.) u + (-u) = 0

5.) c(u + v) = cu + cv

6.) (c + d)u = cu + du

7.) c(du) = (cd)u

8.) 1u = u