I have read over my notes many times but I am still not able to grasp the meaning of span and linear transformation. Can someone please explain them to me?
A set of vectors span some space if the vectors are linearly independent.
The span doesn't need to span the dimension.
If we are working in , our span could be 2 vectors in the xy plane as long as they aren't linearly dependent.
A linear transformation in V maps the a vector v in V to W following some predefined rule.
L(x) = 3x.
All the vectors from V to W are 3 times longer in W.
There are only two operations in a vector space- adding vectors and multiplying vectors by numbers.
The "span" of a set of vectors is the space of all vectors you can get by adding vectors in the set and multiplying by numbers.
A linear tranformation if a function from one vector space to another (or to itself) that "plays nicely" with those operations: F(u+ v)= F(u)+ F(v) and F(av)= aF(v) where F is the function, u and v are vectors and a is a number.
The objects v of linear algebra can be represented as n-tuples of numbers, (x1,x2,x3,..xn), usually called vectors, and a rule for addition of vectors, v1+v2, and multiplication by a scalar. For n=3:
(x1,x2,x3) + (y1,y2,y3) = (x1+y1, x2+y2, x3+y3)
a(x1,x2,x3) = (ax1,ax2,ax3)
A linear transformation is a rule T which assigns to every vector another vector such that
T(x+y)= Tx + Ty
aTx = Tax
Example: assign to every vector v the vector av.
A linear combination of vectors is defined by:
a1v1+a2v2 + a3v3+ ..... anvn.
The span of a set of vectors is all linear combinations of the vectors.
Ex: All linear combinations of two non-parallel vectors in 3d space is a plane.