chapter2.1 linear transformation and matrices
chapter 2.1 Linear Transformations of Euclidean Spaces(선형변환함수)
linear system of the form Ax = b with an m x n matrix A
->we could regard Ax as a function and view Ax = b as meaning that the function maps the vector x to the vector b.
=> f(x) = Ax with an m x n matrix A
the domain(정의역) of this function should be a subset of Rn, the codomain(공역) be a subset of Rm
f(av1 + bv2) = (av1 + bv2) = aAv1 + bAv2 =af(v1)+bf(v2)
->such functions of the for Ax preserve the linear combinations
=> The functions in a vector space that preserve the linear combinations=linear transformations
A function T : Rn -> Rm 은 두 조건들을 만족해야 a linear transformation로 볼 수 있다.
Rn에 속하는 all vectors u, v , all scalars r.
linear transformations(선형변환)은 linear combination(선형결합)을 보존한다.
이 특성은 선형 변환이 공역에서 정의역의 일부 대수 구조를 보존한다는 것을 보여줍니다.
a linear transformation of R into R is completely determined by the value of T(1)
a linear transformation T : Rn -> Rm is determined by its values on a basis for Rn
T : Rn -> Rm be a linear transformation, and let B = {b1, . . . , bn} be a basis for Rn
v ∈ Rn에서,the vector T(v) is uniquely determined by the vectors T(b1), . . . ,T(bn)
the unique linear combination of b1, . . . , bn so that v = r1b1+· · ·+rnbn
T is a linear transformation->
{e1, . . . , en} be the standard basis for Rn이고 T : Rn -> Rm 는 a linear transformation라 두자
the matrix A = [T(e1), . . . ,T(en)]
A : jth column is T(ej) 을 갖는 m x n matrix ,T(x) = A(x) ( x ∈ Rn)
