• The identity map id(v ) = v and the zero map 0(v ) = 0 are linear transformations. It is easy to see that a linear transformation ϕ : V → W is uniquely determined by the values of f on a basis in V . Indeed we have the following definition. Definition 2. Let V and W be vector spaces over F and B be a basis of V . Any arbitrary function f : B → W can be extended by linearity to a unique transformation ϕ : V → W , that is ϕ(λ1 x1 + · · · + λk xk ) = λ1 f (x1 ) + · · · + λk f (xk ) for any x1 , . . . , xk ∈ B and λ1 , . . . , λk ∈ F. There is a one-to-one correspondence between a function from B to W and a linear map from V to W . 2
算法
1.3
Arithmetics on linear maps
Definition 4. Let f, g : V → W be linear transformations and λ ∈ F . The sum of f and g is the linear transformation f + g : V → W defined by (f + g )(x) = f (x) + g (x). The scalar multiplication of f and λ is the linear transformation λf : V → W defined by (λf )(x) = λf (x). Definition 5. Let f : V → W and g : U → V be linear transformations. The composition of f and g is the linear transformation f ◦ g : U → W defined by (f ◦ g )(x) = f (g (x)). Definition 6. Let f : V → W be an isomorphism. The inverse of f is the linear transformation f −1 : W → V defined by f −1 (x) = y whenever f (y ) = x. Examples: • Let A, B ∈ Mm,n (F ). Consider f, g : F n → F m defined by f (x) = Ax and g (x) = Bx. Then (f + g )(x) = (A + B )x and (λf )(x) = λAx. • Let A ∈ Mm,n (F ) and B ∈ Mn,p (F ). Consider f : F n → F m defined by f (x) = Ax and g : F p → F n defined by g (x) = Bx. Then (f ◦ g )(x) = ABx. • Let A be an invertible n × n matrix. Consider f : F n → F n defined by f (x) = Ax. Then (f −1 )(x) = A−1 x. 4
2
2.1
Matrix of transformation
Definition
Definition 7. Let ϕ : V → W be a linear transformation, B = (x1 , . . . , xn ) be an ordered basis of V and C = (y1 , . . . , ym ) be an ordered basis of W . The matrix of the linear transformation of ϕ relative to B and C is MCB (ϕ) = ([ϕ(x1 )]C [ϕ(x2 )]C · · · [ϕ(xn )]C ). Examples: x x+y = . y x−y Two possible ordered bases for R2 are B = (e1 , e2 ) and C = (e1 +e2 , e1 −e2 ). We have Consider f MBB (f ) = 1 1 , MCB (f ) = 1 −1 1 0 , MBC (f ) = 0 1 2 0 , MCC (f ) = 0 2 1 1 . 1 −1
1.2
Image and kernel
Definition 3. Let ϕ : V → W be a linear transformation. The kernel of ϕ, denoted by ker(ϕ) is the set ϕ−1 (0) = {x ∈ V : ϕ(x) = 0}. The image of ϕ, denoted by im(ϕ) is the set ϕ(V ) = {ϕ(x) : x ∈ V }. Theorem 1. The kernel and the image of a linear transformation ϕ : V → W are subspaces of W . Examples: • Let A ∈ Mm,n (F ). Consider f : F n → F m defined by f (x) = Ax. The kernel of f is the null space of A and the image of f is the column space 线性方程组的所有解的集合是A的零空间 of A. • Consider f : Pk → Pk defined by f (p(x)) = xp (x). It kernel is the set of constant polynomials and its image is the set of polynomials with zero constant term. Theorem 2. Let ϕ : V → W be a linear transformation. 1. ϕ is injective if and only if ker(ϕ) = 0.
单射的
函数f被称为是单射时，对每一值域内的y，存在至多一个定义域内的x使得f(x) = y。
2. ϕ is surjective if and only if im(ϕ) = W .
满射的 满射，意思就在满射里，X经过F到Y中时，Y正好都在X中有原像，Y中没有富余或者多出来的像。
Theorem 3. (Dimension theorem) Let ϕ : V → W be a linear transformation, then dim ker(ϕ) + dim im(ϕ) = dim V. Proof. Take a basis B of ker(ϕ) and extend it to a basis B ∪ C of V , then ϕ(C ) is a basis of im(ϕ). Theorem 4. Suppose dim V = dim W . Let ϕ : V → W be a linear transformation. The following statement are equivalent: 1. ϕ is an isomorphism, that is, a bijective linear transformation.
n. 同形
双射的
既是单射又是满射的映射称为特殊双射，亦称“一一双射”
2. ϕ is injective. 3. ϕ is surjective. 3
Examples: Let A ∈ Mm,n (F ). Consider f : F n → F m defined by f (x) = Ax. • f is injective if and only if N ull(A) = 0. • f is surjective if and only if C (A) = F n . • dim N ull(A) + dim C (A) = n. • In the case the m = n, f is bijective if and only if A is invertible if and only if Ax = 0 has only trivial solution if and only if Ax = b is solvable for all b ∈ F n .
1
1
1.1
Linear transformation
Introduction
Definition 1. A linear transformation (or a linear map) is a function ϕ from a F-linear space V to a F-linear space W satisfying • ϕ(v1 + v2 ) = ϕ(v1 ) + ϕ(v2 ) for any v1 , v2 ∈ F and • ϕ(λv ) = λϕ(v ) for any λ ∈ F, v ∈ V . Examples: • Let A ∈ Mm,n (F). The map f (x) = Ax is a linear transformation from Fn to Fm . • Let A ∈ Mm,n (C). The map f (X ) = XA is a linear transformation from Mp,m (C) to Mp,n (C). • The map A → At is a linear transformation. • The map f : Mm,n (C) → Mn,m (C) such that f (A) = A∗ is a real linear transformation, but not a complex transformation. • Differentiation