a linear map from to is a function with the following properties.
- additivity for all .
- homogeneity for all and all .
some stuff from college
let be vector spaces over the field , and let be a function which would be called a linear map if:
- for every ,
- for every and ,
let be a linear transformation,
consider the transformation of the 2d plane : where this transformation maps the line , where or to the line
use the lines parametric form to find the image of an arbitrary point on the original line, then convert the obtained parametric coordinates of the image into an implicit equation
check whether this represents a linear map:
it isnt, a counter example:
if is a linear map then
if is a function such that then isnt a linear map
if is a linear map then:
- the kernel of is
- the image of is , which means all the values that can be returned by , which is also the column space of its corresponding transformation matrix (if it has one)
given the linear operator defined by
find the matrix of in the basis
on the left columns of the matrix lies the basis and on the right 4 columns lies the image
A=matrix(QQ, [[1,1,0,0, 2,0,1,0], [1,-1,1,0, 2,-1,0,1], [1,0,-1,1, 2,0,-1,0], [1,0,0,-1, 2,1,0,-1]]); fm(r'\[{A} \implies {A.echelon_form()}\]')
and so the matrix is:
fm(r'\[[L]_A^B = {A.echelon_form().submatrix(0,4,4,4)}\]')
find the bases of
finding the basis of
to find the basis of we reduce the matrix:
and so the basis of is and
we apply a simple check:
given the function defined by
prove is a linear transformation
's domain is a vector space of polynomials of the second degree at most,
assume , need to prove
let , need to prove
therefore, indeed is a linear transformation
find the matrix of in standard basis form
find the bases of