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