Q: What is a vector?
short answer: physical quantities that have a magnitude as well as a direction.
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)
4. such that
5. such that
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)
3) or 0 if and only if
eg 1. dot product