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Norm

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In order to develop the geometry of vectors we need to be able to see when they are the same, when they are different, and by how much. To compare, we can measure lengths of vectors and the angle between them.

Length and angle.jpg

More precisely, this is simply the difference between the vectors, i.e., another vector, and its length is the measure of the difference between them.

Difference between vectors.jpg

$$‖PQ‖ = |OP| - |OQ|.$$

The length of a vector is called the norm.

Compare to the 1-dimensional case:

  • In $ℝ$: vector $v$ is a number, so $‖v‖ = |v|$ is the absolute value.
  • In $ℝ^2$: $v$ is a pair of numbers $(a, b)$ then
    • $‖v‖ = (a^2 + b^2)^{1/2}$.

This is a Pythagorean theorem...

Pythagorean theorem in R2 and R3.jpg

  • In $ℝ^3$: what is the norm of $v$, which is a triple of numbers, $‖v‖ =$?

Using the Pythagorean Theorem three times:

  • $d^2 = a^2 + b^2$,
  • $‖v‖^2 = c^2 + d^2$, so
  • $‖v‖^2 = c^2 + a^2 + b^2$.

In $ℝ^n$: when $v = (x^1, ..., x^n)$, what is the norm of $v$?

Definition. The norm of $v∈ℝ^n$ is $$‖v‖ = (X_1^2+X_2^2+...X_n^2)^{1/2}.$$

Properties.

  • (1) $‖v‖ ≥ 0$, $‖v‖ = 0$, if $v = 0$: Positivity;
  • (2) $‖αv‖ = |α| \cdot ‖v‖$ for any $α∈ℝ$: Homogeneity;
  • (3) $‖u + v‖ ≤ ‖u‖ + ‖v‖$: Triangle Inequality.

The properties are more important than the definition or the formula as it's often the case.

Example.

  • $ ‖(1,1,1,1....1)‖$, here 1 repeated $n$ times, so
  • $= (1^2 +..+ 1^2)^{1/2}$ repeated $n$ times,
  • $= √n$.

Example.

  • $‖(1,1,1)‖ = √3$;
  • $‖(1/√3,1/√3,1/√3)‖ = ‖1/√3(1,1,1)‖^2 = 1/√3‖(1,1,1)‖ = 1/√3×√3 = 1$, a unit vector;
  • $(1,0,...0)$ unit vector in $ℝ^n$;
  • $(0,1,0..0)$ unit vector in $ℝ^n$;
  • $(1/√n,1/√n,...,1/√n)$ unit vector in $ℝ^n$.

Given a vector v ≠ 0, then $$u = v/‖v‖$$ is a unit vector.

Based on the formula for the norm, we have the distance formula for $ℝ^n$ $$d(x, y) = ‖x - y‖.$$

The properties are more important than the formula:

  • 1. $‖x‖ ≥ 0, ‖X‖ = 0$ iff $x = 0$;
  • 2. $‖αx‖ = ‖α‖ \cdot ‖x‖$;
  • 3. $‖x + y‖ ≤ ‖x‖ + ‖y‖$.

(So much for "The whole is larger than the sum of the parts", NOT!)

Works for $$‖X‖ = (X_1^2+X_2^2+...X_n^2)^{1/2}.$$ But also for $$‖X‖ = |X_1|+ |X_2|+...+|X_n|.$$ This is indeed a norm: $$‖X + Y‖ = |X_1+Y_1|+|X_2+Y_2|+...+|X_n+Y_n|$$ $$≤ |X_1|+|Y_1|+|X_2|+|Y_2|+...+|X_n|+|Y_n|$$ $$= ‖X‖ + ‖Y‖.$$

Manhattan metric or norm:

Manhattan norm.jpg

To find the distance between the vectors go along the grid $$‖X - Y‖ = |X_1-Y_1|+|X_2-Y_2|.$$