Introduction To Linear Algebra(1)Linear Equation

Homogeneous Equation

Linear Equation could be written to $Ax=0$, then it could be called homogeneous equation. And there are always a trivial solution $x=0$
(Theorem)
The homogeneous equation $Ax=0$ has solution if and only if the equation has at least one free variable

Nonhomogeneous equation

the solution of nonhomogeneous euqation $Ax=b$ is a set of $w=p+v_h$ where $p$ is an specific solution and $v_h$ is a set of solution of equation $Ax=0$

Linear Independence

Linearly independent : $x_1v_1+x_2v_2+...+x_pv_p=0$ has only trivial solution, then { $v_1,v_2...v_p$} is said to be linearly dependent
Linear dependence relation: this relation among $v_1,....,v_p$ when the $c_1v_1+c_2v_2+..+c_pv_p=0$ weights{ $c_1,c_2,...,c_p$} are not all zero.

Linear Transformation

For transform $T:R^n\rightarrow R^m$, there exists a unique matrix $A$ such that:
$T(x)=Ax$for all $x$ in $R^n$ and matrix $A$ could express as $A=[T(e_1) ...T(e_n)]$ where $e_i$ is $i_{th}$ column of the identity matrix.
one to one mapping $T:R^n \rightarrow R^m$ if and only if each b in $R^m$ image of at most one $x$ in $R^n$
$T$ is one to one transform if and only if $T(x)=0$ has the only trivial solution.