The note of Legendre

Q: What is a vector?

short answer: physical quantities that have a magnitude as well as a direction.
eg.
exteded definition (answer): an element of a vector space.

Q: What is a vector space?

answer: a set whose elements are vectors that obey a certain set of rules.
1. If ,
2. If , then (c: constant)
3.
4. such that
5. such that
6.
7.
8.
9.
10.
eg 0.
eg 1.
cf.

eg 2. V: matrices

eg 3. Laplace equation (→ normal modes)

<basis> a set of linearly independent vectors that spans V. say
forms a basis for V


if and only if a_i = 0
and (any vectors in V)

<inner product>

a mapping that maps two vectors to a scalar. < v_i,v_j >
1) < au₁ + bu₂,v > = a < u₁,v > + b < u₂,v > (a,b are constants)
2)
3) or 0 if and only if
eg 1. dot product



eg 2.

Dot product

Cross product