線性轉移的組合

Def:

if :U->V, :V->W are linear transformation, then composition of T2 with T1, denoted by o, is the function defined by (o)(u) = ((u)).

類似復合函數的定義

Theorem: 線性轉移的復合依然線性

If , are linear, then ( is a linear transformation.

Proof:

證明o(cx+y) = c*o(x) + o(y)即可

o: U->W , ∀x, y ∈ U, c ∈ F

o(cx+y)

= ((cx+y)) ---by def

= (c*(x) + (y)) ---by properties of linear

= (c*(x)) + ((y)) ---by properties of linear

= c*o(x) + o(y) --- dy def

Note:線性轉移與基底轉移與標準矩陣

is linear , , are ordered standard bases for and , respectively.

Then, , by theorem, we have:

,here,

is called the standard matrix of T.

線性轉移必可用矩陣乘法表達，線性轉移又是基底的轉移，一個原本的標準基底經過轉移成為一個新的column vector，這些column vector堆砌成,即線性轉移的矩陣化表達(線性轉移的標準矩陣)。

Theorem: 由標準基底向任意基底推廣的線性轉移的矩陣化定理

Let: be linear and , be ordered bases(not only the standard bases) for and , repectively.

Then is a m-by-n matrix such that

where