Linear Algebra - Vector Vector Operations

Vector Vector Operations:

3 - List

Using operator overloading, you can compute this operations with the following syntax.

Operation Syntax
vector negation -v
vector subtraction u-v
scalar-vector multiplication alpha*v
division of a vector by a scalar v/alpha
dot-product u*v
getting value of an entry v[d]
setting value of an entry v[d] = …
testing vector equality u == v
pretty-printing a vector print(v)
copying a vector v.copy()

4 - Operations

4.1 - Translation

A vector translation is also known as a vector Addition.

4.1.1 - Syntax

[u1, u2, . . . , un] + [v1, v2, . . . , vn] = [u1 + v1, u2 + v2, . . . , un + vn]
v + 0 = v

4.1.2 - Property

(x + y) + z = x + (y + z)
x + y = y + x

4.1.3 - Representation

Addition of vectors over $\mathbb{R}$ can be visualized using arrows.

4.1.4 - Computation

In python:

• For two n-vectors
def addn(v, w): return [v[i]+w[i] for i in range(len(v))]
• For n n-vectors:
>>> vectorList = [[1,2,3],[1,2,3], [1,2,3]]
# How to group the number by position in the vector
>>> [[i[j] for i in vectorList] for j in range(len(vectorList[0])) ]
[[1, 1, 1], [2, 2, 2], [3, 3, 3]]
# Then add a sum to the above statement
>>> [sum([i[j] for i in vectorList]) for j in range(len(vectorList[0])) ]
[3, 6, 9]

4.4 - Element-wise multiplication

Element-wise multiplication is the default method when two NumPy arrays are multiplied together.

The element-wise calculation is as follows: $$\mathbf{x} \odot \mathbf{y} = \begin{bmatrix} x_1 . y_1 \\\ x_2 . y_2 \\\ \vdots \\\ x_n . y_n \end{bmatrix}$$

Example: $$\begin{bmatrix} 1 \\\ 2 \\\ 3 \end{bmatrix} \odot \begin{bmatrix} 4 \\\ 5 \\\ 6 \end{bmatrix} = \begin{bmatrix} 4 \\\ 10 \\\ 18 \end{bmatrix}$$

6 - Cross Property

6.1 - Distributivity

Scalar-vector multiplication distributes over vector addition (translation):

$$\alpha(u+v)=\alpha.u+\alpha.v$$

2([1, 2, 3] + [3, 4, 4]) = 2 [1, 2, 3] + 2 [3, 4, 4] = [2, 4, 6] + [6, 8, 8] = [8, 12, 14]

Addition and scalar multiplication are used to defined a a line that not necessarily go through the origin.