Properties of linear transformation - Linear Transformation

Notes:

Recall from last chapter the 2 properties of $Ax$:

1. $A(u+v)=Au+Av$

2. $A(cu)=c(Au)$

where $u$ and $v$ are vectors in $\Bbb{R}^n$ and $c$ is a scalar.

Now the properties of linear transformation are very similar. Linear transformation preserves the operations of vector addition/subtraction and scalar multiplication. In other words, If T is linear, then:

1. $T(u+v)=T(u)+T(v)$

2. $T(cu)=cT(u)$

3. $T(\vec{0})=\vec{0}$

We can even combine property 1 and 2 to show that:

$T(cu+dv)=cT(u)+dT(v)$
where $u$, $v$ are vectors and $c$, $d$ are scalars. Note that if this equation holds, then it must be linear.

If you have more than 2 vectors and 2 scalars? What if you have p vectors and p scalars? Then we can generalize this equation and say that:

$T(c_1 v_1+c_2 v_2+\cdots+c_p v_p )=c_1 T(v_1 )+c_2 T(v_2 )+\cdots+c_p T(v_p)$
Again if this equation holds, then it must be linear.