Revision as of 06:24, 8 December 2010

MA 265 Chapter Revies... Chapter 1 - Chapter 5

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Chapter 1

Matricies

A = [a b c; d e f; g h i] this is a 3x3 Matrix. 3 rows and three colums. The rows are seperated by a semicolen.

B = [a b c d; e f g h] this is a 2x3 Matrix. 2 row and 4 colums. Entry Asub(2,3) = g

- matricies are only equal if each corrisponding entry is the same.

- Matrix Addition = add the values of the corresponding matrix entries.

- Matrix Scalar Multiplication = r[a1 a2; a3 a4] = [ra1 ra2; ra3 ra4]

- Linear Combinations = c1A1 + c2A2 + ... + ckAk where c is a scalar and A is a matrix.

- Transpose Matrix(A^T) = say A = [a1 a2 a3 a4; b1 b2 b3 b4; c1 c2 c3 c4] the A^T = [a1 b1 c1; a2 b2 c2; a3 b3 c3; a4 b4 c4]

     notice that A   is a (3x4) matrix = 3 rows and 4 columns 
                 A^T is a (4X3) matrix ... the rows and the colums interchange.

Back to 2010 Fall MA 265 Momin

- Matrix Multiplication : C = a1ib1j + a2ib2j + .... + aipbip where the a components are of one matrix and the b components are of another. the i components are also the comonents of the row values of a and the j components are the column components of the matrix b. Remeber: when multiplying matricies one always multiplies row(a) x column(b). Also: if matrix A = (3*5) and B = (5*1) and C= A*B the C = (3*1) or... becuase the matrix A and B are complatible to be multiplied, the resulting matrix C is the row component of A and the column component of B.

- Some Properties

    1.  A(BC) = (AB)C
    2. (A+B)C = AC + BC
    3. C(A+B) = CA + CB

    1. (A^T)^T = A
    2. (A+B)^T = A^T + B^T
    3.  (rA)^T = rA^T

-Diagonal Matrix :for matrix A with components a..... aij = 0 for i not equal j ex. [1 0 0; 0 2 0; 0 0 5]

-Scalar Matrix : Diagonal matrix where all diagonal are equal ex. [4 0 0; 0 4 0; 0 0 4]

-Identity Matrix(In) : Diagonal matrix with all ones ex. [1 0 0; 0 1 0; 0 0 1] for A = (3x3) ex. [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] for A = (4x4)

  A*In = A

-Upper and Lower Triangle : are a Symmertric Matrix(S) where A^T = A and/or aij = bij

-Skew Summetrix Matrix(Ak): if A^T = -A and/or aij=-bij Also: entries on Main Diagonal = 0, A=S+K

-Nonsingular : the matrix is invertible and has an inverse. (AB = BA = In)

-Singular : the matrix is not invertible and has no inverse.

 For Singular
  Ax = b         x = A^-1b   where A^-1 is the inverse of A
  (AB)^-1 = A^-1 B^-1
  (A^-1)^-1 = A
  (A^T)(A^-1)T = In

Chapter 2

Row Operation; Reduced Row Echelon Form; Solving Linear Systems

-Reduced Echelon Form (REF):

    1) All zero rows are at the bottom of matrix
    2) First non-zero entry is 1. (leading 1 in rows)
    3) Leading 1 in following rows will appear to the right and below for the preceeding rows.

-Reduced Row Echelon Form (RREF):

    1) All zero rows are at the bottom of matrix
    2) First non-zero entry is 1. (leading 1 in rows)
    3) Leading 1 in following rows will appear to the right and below for the preceeding rows.
    4) If a column contains a leading 1, all other entries of the column must be zeros.
   Note: RREF is exactly the same as REF with the addition of step 4.

-Elementary Row Operations :

    1) One can add one row to the another
    2) One can multiply rows by any scalar
    3) Basically one tries to cancel other rows out by applying any combination of the two operations stated in step 1 and 2.

-Therom 2.1 : Every non zero matrix is row equivalent to a matrix in REF. -Therom 2.2 : Every non zero matrix is row equivalent to a unique matrix in RREF.

-Solving Linear Systems : put matrix into REF or RREF and try to solve the linear system of each equivalent row. This answer will correspond to a variable which each column of the REF or RREF represents.

-Inverse of a Matrix (A^-1) : Make the matrix A an augmented matrix with its Identity Matrix. Then try to make the side that has matrix A into the identity matrix by using row operators and putting the augmented matrix into RREF form. Once the left side of the augmented matrix equals the identity matrix, the right side of the augmented matrix will be the inverse of matrix A or (A^-1). ex. if A = [x1 x1 x3; y1 y2 y3; z1 z2 z3] the the augmented matrix will look like this [x1 x1 x3; y1 y2 y3; z1 z2 z3| 1 0 0 ; 0 1 0; 0 0 1] then one will use row operators to make the new matrix look like this: [1 0 0; 0 1 0; 0 0 1| a1 a2 a3; b1 b2 b3; c1 c2 c3] the Inverse matrix of A; A^-1 = [a1 a2 a3; b1 b2 b3; c1 c2 c3]

-Therom 2.10 : if A = (nxn); or a square matrix; then it is sinuglar if it is row equivalent to matrix B which has a row of zeros. (a non correct way but a different way to think about is that RREF of A has a row of zeros)

-Therom 2.11 : If A and B are both (nxn) matricies, then AB=In and B=A^-1. B is equivalent to A if we can derive B from applying finite row operations to A.

-Therom 2.13 : Two (mxn) matricies are equivalent iff (iff means if and only if) B = PAQ where Q and Q are nonsingular matrices.

-Therom 2.14 : matirx A (nxn) is nonsingular iff A is equivalent to In.