Def of Linear transformation:

Let and be vector spaces(over ). A function is called a linear transformation from into if for all and such that:

<a>

<b>

矩陣乘法是一種特殊的線性轉移

Properties:

  1. is linear

Proof of <1>:

Proof of <2>:

重要：Theorem(線性轉移必可矩陣化且可視為基底之轉移):

Let be linear and be the standard basis for , then there exist a m-by-n matrix such that where

proof:

then we have:

, substitute it into

i.e.

(By(3))

where

Note:

  2. Here, is called the standard matrix for and and is called multiplication by A.