# The note of Legendre

to the "Terra incognita"

## Mathematical Physics 1

2007. 9. 3. 20:17

### Q: What is a vector?

short answer: physical quantities that have a magnitude as well as a direction.
eg.

r, v, a in vector

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 ai = 0
and
(any vectors in V)

## <inner product>

a mapping that maps two vectors to a scalar. < vi,vj >
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.

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