Linear Algebra: Inverse Transformation

1. Inverse Transformation
Constant transformation, defined as IX: Rn-> RM, in (x) = x; equivalent to transformation from itself to itself;
Define the inverse conversion, and convert FX: Rn-> RM. If f'y: Rm-> RN, f' of = in and fof' = IM, f' is called the inverse transformation of F, they can also be called inverse functions. These two conditions can actually prove each other. We can also see from the form that F and F' are reciprocal functions.
F has an inverse function. It can be said that F is reversible.

F is reversible, which also means f (x) = V. The equation has a unique solution.

2. Full shot and single shot
If a function F: X-> Y, for each member y of Y, at least one X satisfies f (x) = Y, then function f is a full-Shot function;
If each member of Y has a maximum of one X that satisfies f (x) = Y, then function f is a single-shot function.
If a function is reversible, it must be a full shot and a single shot.

3. How to determine whether the transformation is reversible
Transformation can be represented by matrix multiplication AX = Y. A is an mxn matrix;
First, it must be a full shot, so any vector of the Vector Space RM must be expressed as ax, then the column space of a = RM, so the rank of a = m;
Secondly, it must be a single shot. For AX = Y, each valid y value has a unique solution to the equation, so the zero space of a can only contain one vector: zero vector. In this case, the column vector of A is linearly independent, so the rank of a is = n.

Proof of the second point: Assume that a solution of Ax = B is interpreted as X, and the space of the Solution of AX = 0 contains not only the zero vector, then x + Xn (XN is the non-zero vector of the zero space) it is also an effective solution of Ax = B. In this way, Ax = B has a non-unique solution.

In this case, M = n, and A is a matrix.

4. inverse transformation is also a linear transformation.
Assuming that T is a linear transformation and reversible, t's inverse transformation T' is also a linear transformation: t' (a + B) = t' ot (t' () + t' (B) = t' (tot' (A) + tot' (B) = t' (A + B );
T' (CA) = t' ot (t' (CA) = t' (ctot '(a) = t' (T (CT' ())) = CT '().

5. Matrix reciprocal of transformation and Inverse Transformation
Tot' = A * a' = in; the matrix of constant transformation must be a matrix of units. Therefore, a and A' are reciprocal matrices.

6. Method for Finding the inverse matrix
You can use matrix A to simplify the step ladder to obtain the inverse matrix. Using the n-step transformation, A is transformed into a simplified step ladder and further transformed into a matrix of units. Each step transformation is equivalent to a linear transformation of all column vectors of, this transformation can be represented by the transformation matrix Si. There are s1s2... SNA = I =, s1s2... Sn = '.

Therefore, when we perform a row transformation on a, we perform the same Transformation on I. When a changes to the unit matrix, I changes to '.

7. Inverse Of The 2x2 Matrix
Two-dimensional matrix [{A, c}, {B, d}]. Based on the above method, we can obtain its inverse matrix (1/(Ad-BC) [{d, -c}, {-B, A}].
As long as Ad-BC is not zero, the matrix is reversible.
Ad-BC is also called the determining factor of a matrix.