2015.11.23---2015.11.28 linear algebra

This week has focused on a number of topics, including:

1) from the perceptual knowledge of what is linear algebra, and from the ideological realization that linear algebra is useful.

2) Simple understanding of the addition, subtraction, multiplication of the matrix. These are some of the rules that people prescribe. Mastery can be.

3) Inverse of the matrix, from the basic method (cofactor-type → algebraic cofactor → adjoint matrix) and Gaussian method to solve the inverse of the matrix.

4) Several applications related to matrices are discussed, including the expression of linear equations and vectors, and from which we learn that the essence of abstract problems from different questions is the same mathematical problem.

5) If the inverse of a matrix does not exist or is undefined, the reason is | a| = 0. Inverse of a = 1/| a| Adj (A). From a mathematical point of view, it is the case that a linear equation is parallel or the matrix is a common-line vector.

6) Then we solve a ternary three-order equation by the upper triangular determinant. Learned to solve the simplest 3-dollar linear equations.

7) We have learned some basic concepts of vectors, including the most basic addition and subtraction of vectors. Understand the field of real numbers, denoted by R, and R^n is a dimension problem. Some of the operations of vectors are also defined as some of the rules, first appeared in Newton's writings, and from the real world we have also given some reasonable explanations,

8) from the angle of the vector, the problem of solving the linear equation by two points is solved in a more profound way in geometry. At the same time, it is understood that in three-dimensional or higher-dimensional systems, to describe the line, can only be described from the parametric equation.

9) Then, through an arbitrary algebraic combination between two vectors that are not collinear, we elicit the span (), and the two non-collinear x1,x2 can span out the r^2 plane. If the two vectors are collinear, we say they are linearly correlated, and we also understand that the equations consisting of simple multiplication and subtraction are linear.

10) Then we learned how to judge the linear correlation and irrelevant criteria, that is, to see [x1,x2,x3]. If the C1,C2,C3 in c1x1+c2x2+c3x3= 0 is not all zeros, at least one is not 0, then the matrix is linearly correlated.

11) We then introduced a specific example to detect linear correlations.

11) Then we learned to quantum space, knowing the way to judge the Quantum Space: (1) must go through 0 vectors, (2) The logarithmic multiplication is closed, (3) The vector addition is closed.

12) We then cite a few examples to judge the quantum space.

13) Then we introduce the concept of SVM, the meaning of the base existence is that the base can be represented by a linear combination of arbitrary vectors. The vector base must not be collinear, and it has important meanings in physics and other applications.

14) Then we learn the dot product of the vector, the dot product of the vector is a scalar. Then it proves that the Exchange law of dot product, the distribution rate and the binding law are all established.

15) We then prove the Cauchy inequality, which is proved by the constructor p (t) = |t y-x |^2. The theorem is |x|+|y| >= |x + y | When, and only if, x = cy, the equation is set.

16) We then prove the triangular inequalities. Theorem is |x + y | <= |x| +| y|

17) We then draw the definition of the vector angle, which is introduced by the analogy of a geometric triangle. The expression of the vector angle is obtained by cosine theorem, cosθ= a b/|a| | b|

18) by introducing the concept of vector angle, we draw the method of judging vector perpendicular. If A. b = 0. Then two vectors are intersecting and perpendicular. If two non-zero vector A. b = 0, then the two vectors are perpendicular.

November 28, 2015 21:45 written at Sichuan Polytechnic Institute Isian first experimental building

2015.11.23---2015.11.28 linear algebra