linear transformation
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
- 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:
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
, is define as
- is that image is a part of the range?
- yes, when S is V, we called the image as range.
- range codomain
- the def. for any element in A 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
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
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