linear map

a linear map from $\text{[math]}$ to $\text{[math]}$ is a function $\text{[math]}$ with the following properties.

additivity. $\text{[math]}$ for all $\text{[math]}$ .
homogeneity. $\text{[math]}$ for all $\text{[math]}$ and all $\text{[math]}$ .

[cite:;taken from @algebra_axler_2024 chapter 3 linear maps; definition 3.1 linear map]

any $\text{[math]}$ matrix $\text{[math]}$ defines a linear transformation $\text{[math]}$ by matrix multiplication: $\text{[math]}$ 2. every linear transformation $\text{[math]}$ is given by multiplication by the $\text{[math]}$ matrix $\text{[math]}$ : $\text{[math]}$ where the $\text{[math]}$ th column of $\text{[math]}$ is $\text{[math]}$ .

[cite:;taken from @calc_hubbard_2015 theorem 1.3.4]

some stuff from college

let $\text{[math]}$ be vector spaces over the field $\text{[math]}$ , and let $\text{[math]}$ be a function which would be called a linear map if:

for every $\text{[math]}$ , $\text{[math]}$
for every $\text{[math]}$ and $\text{[math]}$ , $\text{[math]}$

let $\text{[math]}$ be a linear transformation, $\text{[math]}$

consider the transformation of the 2d plane $\text{[math]}$ :
$\text{[math]}$ where
$\text{[math]}$ this transformation maps the line $\text{[math]}$ , where $\text{[math]}$ or $\text{[math]}$ to the line
$\text{[math]}$

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

$\text{[math]}$ is $\text{[math]}$ a linear map?

we need to prove $\text{[math]}$

i should be writing $\text{[math]}$ not $\text{[math]}$

$\text{[math]}$

we need to prove $\text{[math]}$
$\text{[math]}$

check whether this represents a linear map:
$\text{[math]}$

it isnt, a counter example:
$\text{[math]}$

if $\text{[math]}$ is a linear map then $\text{[math]}$

if $\text{[math]}$ is a function such that $\text{[math]}$ then $\text{[math]}$ isnt a linear map

if $\text{[math]}$ is a linear map then:

the kernel of $\text{[math]}$ is $\text{[math]}$
the image of $\text{[math]}$ is $\text{[math]}$ , which means all the values that can be returned by $\text{[math]}$ , which is also the column space of its corresponding transformation matrix (if it has one)

given the linear operator $\text{[math]}$ defined by $\text{[math]}$

find the matrix of $\text{[math]}$ in the basis $\text{[math]}$

$\text{[math]}$ on the left columns of the matrix lies the basis and on the right 4 columns lies the image
src_sage{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' $\text{[math]}$ ')} {{{results( $\text{[math]}$ )}}}
and so the matrix is:
src_sage{fm(r' $\text{[math]}$ ')} {{{results( $\text{[math]}$ )}}}

find the bases of $\text{[math]}$

finding the basis of $\text{[math]}$
$\text{[math]}$

to find the basis of $\text{[math]}$ we reduce the matrix:
$\text{[math]}$ and so the basis of $\text{[math]}$ is $\text{[math]}$ and $\text{[math]}$
we apply a simple check: $\text{[math]}$