Linear Algebra (1) (linear algobra with Application) Steven J. Leon (Eighth Edition)

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• Exercise: (use MATLAB whenever possible)
• Exercise:

I started learning linear algebra for the exam today. Although I have learned it before, I have forgotten it in my family.

Chapter 1 matrix and equations 1.1 Linear Equations

The content of this section is very simple, that is, a brief introduction of things to be learned.

Definition: If the coefficient of the first K-1 variable of the K equation in the equations is 0 and the coefficient of XK is not 0, it is called strict triangular form)

Exercise: (use MATLAB whenever possible)

1. (a) (B) (c) (d)

>> A = [1,-3;0,2];>> x = A\[2;6]

>> A = [1,1,1;0,2,1;0,0,3];>> x = A\[8;5;9]

>> A = [1,2,2,1;0,3,1,-2;0,0,-1,2;0,0,0,4];>> x = A\[5;1;-1;4]

 >> A = [1,1,1,1,1;0,2,1,-2,1;0,0,4,1,-2;0,0,0,1,-3;0,0,0,0,2];>> x = A\[5;1;1;0;2]

Other questions are the same as this, so they will not be written.

1.2 step ladder

Definition: If a Matrix meets

(1) The first non-zero element in each non-zero row is 1

(2) When the yuan in the k-th row is not all 0, the number of the first 0 of the first variable in the k + 1 is more than K rows.

(3) All rows whose elements are 0 must be behind the rows whose elements are not all 0.

This kind of matrix is called row Echelon form)

Understanding over-definite equations and subdefinite Equations

Hyperdefinite equations are usually incompatible -------- because there are too many conditions (number of equations), -------- the number of equations in the equations is more than the number of unknown variables

Subdefinite equations are generally compatible ---------- the number of equations is smaller than the number of variables --------- (M <n)

Definition: If a Matrix meets

(1) matrix type step-wise

(2) The first non-0 element of each row is the unique non-0 element of the column.

In this matrix, we call the short for rRef

Exercise:

1.1

A =     1     2     3     4     0     0     1     2>> rref(A)

Other similar

1.3 matrix Arithmetic

Mark the matrix as a = (AIJ) (indicating that I can mark the subscript in red, but it cannot be typed)

The concept of vector is so red that the matrix of 1xn is called row vector. The matrix of Nx1 is called column vector)

A set of solid matrices of all column vectors is called the n-dimensional Euclidean space (Euclidean N-space ).RN (blue indicates the upper mark)

1.4 matrix algebra

Unit Matrix, non-singular matrix (reversible matrix), multiplication inverse element (multiplicative inverse)

A matrix does not have an inverse matrix. We say it is a singular matrix. Whether a matrix is a singular matrix is for a matrix. Only a matrix can be determined. others cannot.

1.5 Elementary Matrix

Elementary Matrix-only one elementary row operation starts from the unit matrix I. The obtained matrix is the elementary matrix.

There are three primary Matrices

(1) The elementary matrix obtained from the two rows in the Exchange Matrix

(2) Multiply a row in the matrix of units by a constant. The matrix obtained is an elementary matrix.

(3) adding a row in the Matrix to a row by multiplying a constant is an elementary matrix.

Diagonal Matrix and Triangle Matrix

Upper Triangular and lower triangular)

Triangle Decomposition

The angle element of matrix L is a 1 lower Triangle Matrix. We call it a unit lower Triangle Matrix, the process of dividing matrix A into the product of a unit lower Triangle Matrix and a strict upper Triangle Matrix U is usually called Lu factorization)

1.6 Matrix Blocks

Four scenarios

Expand Outer Product