 # 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.    ### Dot product    ### Cross product      , , , , , , , , , , , ,

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