Chap 1: Linear Equations and Matrix

• Form of floating number: f=± .d1 d2 … dt * b^n (d1≠0);

• Roundoff error: caused by the different magnitudes between the different columns;

• Partial pivoting: search the position BELOW the pivotal position for the coefficient in maximum magnitude;

• Complete pivoting: search the position BELOW and on the RIGHT of the pivotal position for the max coefficient;

• Partial & Complete pivoting: whether using elementary column operation. The partial one is used more frequently because the elementary column operation is not easy to use;

• the solution of an ill-conditioned system is extremely sensitive to a small perturbation on the coefficients;

• Geometrical view: two linear systems are almost parallel so that their cross point will move sensitively when any one system moved;

• How to notice the ill-condition of a linear system: enumerating ( it's not easy to find whether a system is ill-conditioned);

• 2 way to solve the problem: bite the bullet and compute the accurate solution, or redesign the experiment setup to avoid producing the ill-conditioned system. The latter one is better empirically. Finding a system is an ill-conditioned one as early as possible will save much time;

• Notation: E;

• Cause: linear correlation between different column vectors and modified Gaussian elimination;

• The echelon form (namely the position of pivots) is uniquely determined by the entries in A. However, the entries in E is not uniquely determined by A.

• Basic column: the columns in A which contain the pivotal position;

• Rank: the number of pivots = the number of nonzero rows in E = the number of basic columns in A;

• Reduced row echelon form: produced by Gaussian-Jordan Method( [0 0 1 0]T ), notated by EA;

• Both form and entries of EA is uniquely determined by A;

• EA can show the hidden relationships among the different columns of A;

• A system is consistent if it has at least one solution. Otherwise, it is inconsistent.

• When n (the number of equations) is two or three, the consistency of the system can be shown geometrically, the common point.

• If  n>3, we can judge through the following method:

• In the augmented matrix [A|b], 0=a≠0 does not exist;

• In [A|b], b is the nonbasic column;

• rank(A|b)=rank(A);

• b is the combination of the basic column in A.

• Homogeneous and nonhomogeneous;

• Trivial solution;

• A homogeneous system must be a consistent system;

• General solution: basic variable, free variable;

Chap 2: Matrix Algebra

• Elementary matrix: I-uv^T, u and v are column vectors;

  • The inversion of an elementary matrix is also an elementary matrix;
  • Elementary matrices associated with three types of elementary row (or column) operation;
  • A is a nonsingular matrix <=> A is the product of elementary matrices of Type I, II and III row (or column) operation;

• Equivalence: A~B <=> PAQ=B for nonsingular P and Q;

• Row equivalence and column equivalence;

• Rank normal form: if A is an m*n  matrix such that rank(A)=r, then A~Nr=[[Ir, 0]^T, [0, 0]^T], Nr is called rank normal form for A;

• A~B <=> rank(A)=rank(B);

• Corollary: Multiplication by nonsingular matrices cannot change rank;

• rank(A^T)=rank(A);

• rank(A*)=rank(A);

Vector Spaces