Linear transformations

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有關於線性轉換的一些筆記，放在這裡以備不時之需

• A transformation :
  • maps in to in
• T is a linear transformation if
• Some linear transformation example:
  • Rotation in
  • Reflection in
• Some instresting transformation(but not linear!)
  • Translation
  • Length
• Question: Is any a linear transformation?
  • Let's check it by definition
    • We know
    • $T(cv) = A(cv) = cA(v) = cT(v)
  • The answer is: Yes
• Composition of two trasformations:
  • It's possible to combine multiple linear transformation to one.
  • If and are linear，their combination is also linear.
  • Combination order matters， not always equal to

Language of transformations

• is a transformation
  • : domain, : codomain
  • : the image of v
  • kernel of = the set of in that
  • Range of = = the set of all images
  • If is a linear transformation
    • We'll try to prove that by linear transformation's definition
    • is a subspace of
      • if , then
      • if , then
    • is a subspace of
      • if , , then
      • if , then
  • = range of , Rank of =
  • Nullify of =
• One to one
  • The linear transformation is one to one if for any
• Onto
  • A linear transformation is onto if
• Isomorphism(同形)
  • A one-to-one linear transformation is onto is an isomorphism
• Linear trans. V.S. Matrix calculation
  • Suppose , is
  • = the set of 's that = Nullspace of =
  • = the set of all images = column space of =
  • == ==
• Recall Rank theorem
  • is one-to-one = = is full column rank(only one solution)
  • T is onto = = = is full row rank(has solution)
  • T is isomorphism
    • A is full rank

Coordinate systems and general vector spaces

• Recall unique representation theorem
  • there exists one and only one way to express any in as a linear combination of
  • Coordinate vecotr in the basis :
  • Let's suppose a basis ，then we want to map a vecotr (which is within ) to ，and and are isomorphism
    • We can achive that by:
• General vector spaces(optional)
  • With defined vector addition and scalar multiplication
  3. exist a unique 0 satisfied
  4. for any , there exists ,

Matrices of linear transformations

graph

• Suppose we have a vector in space
  • Let's say is our input vector
  • is space 's basis
• is the Input coordinate vector
  • i.e.
• Then,
  • is the basis of
  • is our output vector
  • i.e.
• One question is: Is there exist a matirx Satisfied ? (The red path in our figure)
  • 我們可以回想一下定義
  • Input vector
    • 把係數提出來就是, 所以
  • 則Output vector
    • 不要忘記線性轉換的基本定義!
  • 而我們稱呼原本在基底空間中的座標向量為
    • 為中的基底向量, 為中的係數
    • 因此
  • 我們可以思考一下，空間的基底向量經過轉換成中的向量後，一定可以被空間的基底組合出來
    • 組合所使用的係數就可以說它是A矩陣的第一列(col)，把每一列係數放到一個矩陣裡，就是A矩陣!
  • 所以結論就是:
  • 所以圖表中的紅色路線的確存在!