basic

  • input a vector, output a vector
  • it is a type of transformation, so think it in moving way
    • the origin should not change, or it will become “affine transformation”(仿射变换)
    • the grid should keep parallel and evenly distributed 【平行且等距分布】after the transformation, so that can be called as “linear”
    • transformation do strech, spin(trigonometric), projection(object, light, shadow), change the dimension(not invertible)
  • here, the transformation is done by matrix
    • matrix is the operater for the basis
    • matrix stored the ratio! between the basis of VS.VS.
    • because matrix can store the info of linear combination
    • and the linear combination make sure the grid is linear after the transformation, because the the new basis is a result of a linear combination of the old basis
  • since it is linear, we keep the same rule for scalar multiplication and addition
  • 什么是线性变换【吞吐向量的方程】以及如何用矩阵表示【记录了基底变化后的坐标】

Determine linear transformation

  • linear trans. should inheritage the properties from the original basis, implicitly, the new basis fufill all the requrement for a vector space generating set.
  • there are 3 requirements

kernel(矩阵的核)

L is linear trans. from V to W, kernel L is denote as:

ker(L)={x∈V:Lx=0w}ker(L) = \{x \in V: Lx = 0_w\}

here, Lx can also write as L(x) since matrix is operator
(fnc.)

image(矩阵的像)

S is subspace of V and the image of S is denote as
L(S)L(S), is define as

L(S)={L(x):x∈S}L(S) = \{L(x): x \in S\}

  • is that image is a part of the range?
  • yes, when S is V, we called the image as range.
  • range ∈\in codomain
    • the def. for any element in A ∃\exist unique element in B, then B is codomain
    • def of range is, all L(A) in B

Theo 17.12 in slide 17

Similar

Theorem 18.4 (Similarity Result in general vector space)
Let

E={v1,v2,⋅⋅⋅,vn} and F={w1,w2,⋅⋅⋅,wn}E = \{v_1, v_2, · · · , v_n\} \text{ and } F = \{w_1, w_2, · · · , w_n\}

are two ordered bases for a vector space V and L be a linear transformation from V to V.

Let S be the transition matrix corresponding to the coordinate change from F to E.

If A is the matrix representation of L w.r.t. E (taking E as the basis for both domain and co-domain)

B is the matrix representation of L w.r.t. F (taking F as the basis for both domain and co-domain)
then
B=S−1ASB = S^{−1}AS

Definition 18.5 (Similar) Let A and B are two n × n matrices, B is
said to be similar to A if there exists a nonsingular matrix S such that
B=S−1AS.B = S^{-1}AS.