13 Oct 2014

# Linear Algebra@Chapter 2

Tips before start:

- is identity matrix with 2x2 size, i.e.

- matrix A is n x m, matrix B is a x b. AB has size n x b, m must equal a, since A is coefficient matrix, B is vector of variables, each variables get one coefficient (a column), so # of column in A = # of row in B.
- # of row in A is # of equation in the system, # of column in B is the # of column in the final matrix since each Ab will sum to one column

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Multiplication of Matrix

First at all, understand

Then, you can understand

only can sum to one column, is coefficient matrix, is the vector of variables, can sum to one column if the linear system is completed, i.e . Different column in means different linear system for , so will not sum with .

Finally,

Power of Matrix

Transpose of matrix

Theorem

For d. please be reminded that

matrix A, n x

mand matrix B,ax b, gives out AB with m x a,therefore , b x

aand ,mx n , gives out with m x au , v both in ,

and result in 1x1 matrix , called inner product

and result in n x n matrix, called outer product

Inverse of matrix

if , if , then A is singular, vice versa.

Determinants

, if is x , then is unique.