Linear Algebra: linear transformation

1. Vector Transformation
Vector transformation is a function from one vector space to another (or the same) vector space.

In the vector world, this function is called transformation, which is usually represented by the symbol t.

2. linear transformation
For a vector Transformation: T: Rm-> RN, if T (a + B) = T (A) + T (B) and T (CA) are satisfied) = CT (A), transformation T is linear transformation.

3. matrix vector multiplication
Based on the amn matrix, the transformation T (x) = AX can be defined, and T is a linear transformation of RM-> RN.

4. Matrix Representation vector Transformation
Any linear transformation can be expressed in the form of matrix vector multiplication.
For T (x) transformation, if X is n-dimensional, then there is a corresponding unit matrix in = [E1, E2,..., En], and EI is a column vector. Matrix vector multiplication IX = e1x1 + e2x3 +... + enxn = X,
So T (x) = T (e1x1 +... + enxn) = T (e1x1) +... + T (enxn) = T (E1) X1 +... + T (en) xn = [T (E1), T (E2 ),..., T (en)] x,
We can see that T (x) can always be expressed as the form of matrix vector multiplication, and the above proof also gives the method for obtaining the matrix.

5. Images of vector subsets under Transformation
Assuming the location vector v0, V1, then the vector set Connecting v0, V1 can be expressed as {v0 + T (V1-V0) | 0 <= T <= 1 },
Linear transformation T (VT) = T (v0 + T (V1-v0) = T (v0) + T (t (V1) -T (v0). This shows that the transformed vector set is actually a line segment from T (v0) to T (V1.
This feature fully demonstrates that when linear transformation is performed on a ry, you only need to perform linear transformation on the relevant vertex. After the transformation, the ry is reconstructed in the same way.

6. The transformed image of the sub-space
The transformed image of the sub-space, that is, the transformed Vector Set = {T (x)}, may prove that this new vector set is also a vector sub-space according to the definition of linear transformation.
In addition, we can see from 4 that this sub-space is actually the column space of the transformation matrix.

7. Transformed Core
For a specific member vector of the transformed value or image, which is mapped to a subset of the defined domain of this vector, you can find it through the matrix equation.
For example, if T (x) = Ax, the space for solving AX = Y is the subset of the defined domain mapped to y.
If y = 0, this subset is called the transformed kernel and is recorded as Kert (T). We can see that the kernel is actually the zero space of.

8. Addition and multiplication of transformations
Transform T (x) = Ax, S (x) = Bx, because the total number can be expressed by the transformation matrix, so there can be (t + S) (x) = T (X) + S (x); t (CX) = (CT) (x), C is a scalar.

9. Reverse and zoom Conversion
Section 4th describes how to calculate the transformation matrix. It can be simpler for reverse and zoom transformations. For example, if you want to reverse the k-th component of a vector, you only need to multiply EK by-1, and scale by EK by the scaling ratio. The matrix corresponding to the reverse scaling transformation must be a diagonal matrix.

10. Rotation Transformation under r2
In the two-dimensional vector space, rotate the vector counterclockwise to an angle. According to the method in section 4th, after () rotating a, it becomes (COSA, Sina); () after rotating a, it becomes (-Sina, cosa ). Get the transformation matrix [(COSA, Sina), (-Sina, cosa)].

11. Rotation Transformation under r3
In 3D vector space, vector rotation must specify another vector. Assuming that the rotation is centered on the X axis, the transformation matrix can be calculated to obtain [(, 0), (0, cosa, Sina), (0,-Sina, cosa)]. The rotation around the Y axis and Z axis can be calculated using a similar method. The rotation around any vector can be converted to multiple rotation combinations around X, Y, and Z axes.

12. Unit Vector
A vector with a length of 1 can be called a unit vector. If a vector is not a unit vector, it can be converted to a same unit vector (1/| v |) v.

13. Introduction to projection
Projection of vector A on another vector B can be understood as the shadow of vector A in the B direction under the light of Vertical B. This projection is obviously a vector, CB (C is a scalar ).
Create a vertical vector from the end point of a to the direction of B. Assume it is l. Then a, CB, l constitute a triangle, L = A-CB, CB. L = 0. C = A. B/| B | square.
B can be first converted into a unit vector, so the projection of a on B is a. B * B.

14. Projection Matrix
The definition of linear transformation proves that the projection transformation described in section 13 is a linear transformation. It can be expressed as a matrix.
Assume that the projection target vector V is the unit proof, and the projection transformation can be expressed as T (x) = x. V * v. The transformation matrix is [(v1.v1, v1v2) (v1.v2, v2.v2)] based on the Calculation Method in section 4th.

15. Composite Transformation
Composite transformation refers to the combination of two linear transformations. The combination value is used to transform the t of a vector X to get the x', and the X is converted to the X' s to the x '', the combination of the two transformations, TOS (x) = x ''.
It can be proved that TOS is still a linear transformation.

So what is the change matrix of TOS? Suppose T (x) = Ax, S (x) = Bx, then TOS = B (ax) = (BA) X.

16. Combination law and allocation rate of Matrix Multiplication
Matrix A, B, C, A (BC) = (AB) C, A (B + C) = AB + AC. The verification process is omitted.