${\mathfrak {L}}(0)=0$

Image of 0 under ${\mathfrak {L}}(.)$ is zero "0" if ${\mathfrak {L}}(.)$ is linear

Example of linear map for $\mathbb {R} ^{n}$ (in a n dimension case)

$A:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$

$x\mapsto y=Ax$

$A\in R^{nxn}$ nxn matrix

Insert Figure

In the case above:

${\mathfrak {D}}(Domain)=\mathbb {R} ^{n}$

${\mathfrak {R}}(Range)=\mathbb {R} ^{n}$

Analyzing a more general where m≠n:

$A:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$

$x\mapsto y=Ax$

$A\in R^{nxn}$

where: n=row and m=columns

y=Ax

Now consider:

y=Ax

where: y=nx1 matrix

Ax=nx1

b=nx1

$M:\mathbb {R} ^{m}\mapsto \mathbb {R} ^{n}$

$x\mapsto y=Ax+b$

Clearly:

$M(0)=b\neq 0\Rightarrow M(.)$

The $\neq$ signal means that $M(.)$ is not homogenic

$M(.)$ is not a linear map, affine map

Example: Rotation followed by translation

$y=Rx+b$

m=n=2

${\begin{Bmatrix}y_{1}\\y_{2}\end{Bmatrix}}={\begin{bmatrix}R_{11}&R_{12}\\R_{21}&R_{22}\end{bmatrix}}{\begin{Bmatrix}x_{1}\\x_{2}\end{Bmatrix}}+{\begin{Bmatrix}b_{1}\\b_{2}\end{Bmatrix}}$

Note: Equivalency $A\equiv B$

"if and only of" $\equiv$ "equivalent to" $\equiv$ "necessary and sufficient condition

$A\Rightarrow B$ A is a sufficient condition for B

$A\Leftarrow B$ B is a sufficient condition for A

A is a necessary condition for B

${\begin{bmatrix}A\Rightarrow B\end{bmatrix}}\Leftrightarrow {\begin{bmatrix}\mathbf {B} \Rightarrow \mathbf {A} \end{bmatrix}}$

Where: B is non B or negation of B

and A is non A or negation of A

1) $(\Rightarrow )$ (sufficient condition) 2) $(\Leftarrow )$ (necessary condition)

Homework

Question: Is div(.) an x of $M:\mathbb {R} ^{m}\to \mathbb {R} ^{n}$ and $M$ linear?

$\mathbb {R} ^{m}$ , $\mathbb {R} ^{n}$ are species of vectors (tensors,column matrix). Div maps, vector fields

vector fields (vector-value function) into a scalar function.

In other words, domain and range of div(.) are function spaces

Transformation of coordinates

Second order linear PDE`s

Coordinates:

$x=\phi ({\bar {x}},{\bar {y}})$

$y=\psi ({\bar {x}},{\bar {y}})$

Linear coordinate transformation

Let consider:

$u=(x,y)=u(\phi ({\bar {x}},{\bar {y}}),y(\psi ({\bar {x}},{\bar {y}}))$

$u=(x,y)=u({\bar {x}},{\bar {y}})$ * abuse of notation by using "u"

$u=(x,y)={\bar {u}}({\bar {x}},{\bar {y}})$ * this is more rigorous notation

Example:

Let: $u(x)=ax+b$

Consider: $x=x({\bar {x}})=sin{\bar {x}}$ where x $\to$ general function or $(\phi ({\bar {x}}))$

$u(x)=u(\phi ({\bar {x}}))=asin{\bar {x}}+b$

$u(x)=u({\bar {x}})={\bar {u}}$

where:

$u({\bar {x}})$ is an abuse of notation

and ${\bar {u}}({\bar {x}})$ is a more rigorous notation

$u_{x}(x,y)={\frac {\partial u}{\partial x\,}}(x,y)=u_{x}(\phi ({\bar {x}},{\bar {y}}),y(\psi ({\bar {x}},{\bar {y}}))$

$={\frac {\partial {\bar {u}}}{\partial x\,}}({\bar {x}},{\bar {y}})={\frac {\partial {\bar {u}}}{\partial {\bar {x}}\,}}{\frac {\partial {\bar {x}}}{\partial x\,}}+{\frac {\partial {\bar {u}}}{\partial {\bar {y}}\,}}{\frac {\partial {\bar {y}}}{\partial x\,}}$

where:

${\frac {\partial {\bar {u}}}{\partial {\bar {x}}\,}}{\frac {\partial {\bar {x}}}{\partial x\,}}={\bar {u}}_{\bar {x}}$

${\frac {\partial {\bar {u}}}{\partial {\bar {y}}\,}}{\frac {\partial {\bar {y}}}{\partial x\,}}={\bar {u}}_{\bar {y}}$

Define:

${\bar {x}}={\bar {x}}(x,y)=\phi (x,y)$

${\bar {y}}={\bar {y}}(x,y)=\psi (x,y)$

$u_{y}(x,y)={\frac {\partial {\bar {u}}}{\partial y\,}}({\bar {x}},{\bar {y}})={\bar {u}}_{\bar {x}}{\frac {\partial {\bar {x}}}{\partial y\,}}+{\bar {u}}_{\bar {y}}{\frac {\partial {\bar {y}}}{\partial y\,}}$

In a matrix form:

$=\partial {\bar {x}}(u)={\begin{bmatrix}{\frac {\partial {\bar {x}}}{\partial x\,}}&{\frac {\partial {\bar {y}}}{\partial x\,}}\end{bmatrix}}{\begin{bmatrix}\partial {\bar {x}}\\\partial {\bar {y}}\end{bmatrix}}({\bar {u}})$

Homework

${\begin{Bmatrix}\partial x\\\partial y\end{Bmatrix}}(u)={\begin{bmatrix}{\frac {\partial {\bar {x}}}{\partial x}}&{\frac {\partial {\bar {y}}}{\partial x}}\\{\frac {\partial {\bar {x}}}{\partial y}}&{\frac {\partial {\bar {y}}}{\partial y}}\end{bmatrix}}{\begin{Bmatrix}\partial {\bar {x}}\\\partial {\bar {y}}\end{Bmatrix}}({\bar {u}})$

${\begin{bmatrix}{\frac {\partial {\bar {x}}}{\partial x}}&{\frac {\partial {\bar {y}}}{\partial x}}\\{\frac {\partial {\bar {x}}}{\partial y}}&{\frac {\partial {\bar {y}}}{\partial y}}\end{bmatrix}}$ This matrix is known as the Jacobian matrix

Note: Easier and more general

$(x_{1},...,x_{n})\to ({\bar {x}}_{1},...,{\bar {x}}_{n})$

Ind. notation: ${\bar {x}}_{i}={\bar {x}}_{i}(x_{1},...,x_{n})$

$\mathbf {J} _{nxn}={\begin{bmatrix}\partial {\bar {x}}_{i}\\\partial {\bar {x}}_{j}\end{bmatrix}}_{nxn}$

where:

i= row index

j=column index