矩阵分析课件 3

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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 transformatioLeabharlann Baidu 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
• 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
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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
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
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.
d dx
is a linear transformation from Pk+1 to Pk .
d is a linear transformation from the space of twice • Differentiation dx differentiable functions to the space of differentiable space.
Matrix Analysis Lecture 3. Linear transformation
10 October 2015
In this lecture, we will discuss linear transformations which is the center of linear algebra. Each linear transformation can be represented by a matrix. Most operations on matrices have a corresponding to operations on linear transformations. A linear transformation ϕ : V → V , if we choose different basis, may be represented by different matrices. Those matrices are said to be similar. If a property of matrices can be preserved by similarity, then it is consider the (algebraic) property of the corresponding linear transformation. From now on, we will only consider finite-dimensional vector spaces.
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.
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 .