Bonus point project for MA265.


MA 265 Chapter Reviews... Chapter 1 - Chapter 5

by Felix Quasniczka

Chapter 1


Matrices

A =

   $ \begin{bmatrix}      a & b & c \\      d & e & f\\      g & h & i \end{bmatrix} $

This is a 3x3 Matrix with three rows and three columns.

B =

   $ \begin{bmatrix}      a & b & c & d\\      e & f & g & h \end{bmatrix} $

This is a 2x4 Matrix. 2 rows and 4 columns. In matrix B, entry $ a_{2\text{3}} $ = g

- matrices are only equal if each corresponding entry is the same.

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

- Matrix Scalar Multiplication:


$ r \times \begin{bmatrix} a & b & c & d\\ e & f & g & h \end{bmatrix} = \begin{bmatrix} ra & rb & rc & rd\\ re & rf & rg & rh \end{bmatrix} $


- Linear Combination: $ c_{\text{1}}A_{\text{1}} + c_{\text{2}}A_{\text{2}} + ... + c_{\text{n}}A_{\text{n}} $ where c is a scalar and A is a matrix

- Transpose Matrix$ (A^T) $:

$ say A = \begin{bmatrix} a1 & a2 & a3 & a4\\ b1 & b2 & b3 & b4\\ c1 & c2 & c3 & c4\end{bmatrix} $

 $  then  A^T = \begin{bmatrix}                a1 & b1 & c1\\                a2 & b2 & c2\\                a3 & b3 & c3\\                a4 & b4 & c4\end{bmatrix} $


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

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- Matrix Multiplication : C = $ A_{1\text{i}}B_{1\text{j}} + A_{2\text{i}}B_{2\text{j}} + ... + A{i\text{i}}B_{i\text{j}} $ where the A components are of one matrix and the B components are of another. The i components are also the row values of A and B and the j components are the column components of the matrix A and B. Remember: when multiplying matrices one always multiplies row(a) x column(b). Also: if matrix A = (3*5) and B = (5*1) and C = A*B then C = (3*1) or... because the matrix A and B are compatible 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 ... $ a_{i\text{j}} $ = 0 for i not equal j Example

$ \begin{bmatrix} 1 & 0 & 0\\                              0 & 2 & 0\\                              0 & 0 & 5\end{bmatrix} $

-Scalar Matrix : Diagonal matrix where all diagonals are equal Example:

$ \begin{bmatrix}                4 & 0 & 0\\                0 & 4 & 0\\                0 & 0 & 4\end{bmatrix} $

-Identity Matrix$ (I{\text{n}}) $ : Diagonal matrix with all ones Example for A = (3x3) matrix A =

$ \begin{bmatrix}                1 & 0 & 0\\                0 & 1 & 0\\                0 & 0 & 1\end{bmatrix}  $

For A = (4x4) matrix A =

$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix} $

  A*$ I{\text{n}} $ = A

-Upper and Lower Triangle : are a Symmetric Matrix(S) where $ A^T $ = A and/or $ a{i\text{j}} = b{i\text{j}} $

-Skew Symmetric Matrix$ (A{\text{k}}) $: if $ A^T = -A $ and/or $ a{i\text{j}}= -b{i\text{j}} $ Also: entries on Main Diagonal = 0, A=S+K

-Non-singular : the matrix is invertible and has an inverse. (AB = BA = In) (det(A) = 0)

-Singular : the matrix is not invertible and has no inverse. (det(A) does not = 0)

Some Properties of Matrices (Non-Singular):

  • If $ Ax = b $ then $ x = A^{-1} b $ where $ A^{-1} $ is the inverse of A
  • $ (AB)^{-1} = B^{-1} A^{-1} $
  • $ (A^{-1})^{-1} = A $
  • $ (A^T)(A^{-1})^T = I_n $

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 preceding 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 preceding 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.

-Theorem 2.1 : Every non zero matrix is row equivalent to a matrix in REF. -Theorem 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 singular 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) matrices, 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) matrices are equivalent iff (iff means if and only if) B = PAQ where Q and Q are non-singular matrices.

-Therom 2.14 : matrix A (nxn) is non-singular iff A is equivalent to In.



Chapter 3


Determinants

-Determinants : Determinants only work for Square Matricis (nxn)

-Permutations : has an inversion if a larger interger preceeds a smaller one.

    even permutation - the total number of inversions are even
    odd permutation - the total number of inversions are odd
    Sign : (-1)^(# of inversions)

-Determinant (det):

  • If

$ A = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} $

then

$ \det(A) = \det \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} = ( (a1*a4) - (a2*a3) ) $

  • If

$ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{bmatrix} $

then

$ \det(A) = [(a1*a5*a9)+(a2*a6*a7)+(a3*a4*a8)]- [(a7*a5*a3)+(a8*a6*a1)+(a9*a4*a2)] $


this is how one calculates the determinant of a (2x2) and (3x3) matrix. Following this result by hand will make it much clearer to understand. For the (3x3) matrix, we took the original matrix and wrote the first two terms at the end of each row. Then we took the sum of the diagonals similar to how we summed the (2x2) matrix. NOTE: This does not apply for a matrix where n >= 4.

-Properties of Determinants :

 1) det(A) = det(A^T)
 2) det(B) = det(A)    If B results from interchanging 2 rows of A.
 3) det (B) = k*det(A) If B is obtained from multiplying a row in A by k.
 4) If two rows/columns are equal, det(A) = 0
 5) If there is a row of zeros det(A) = 0
 6) det(B) = det(A)    If any of the elementary operations are used on A to obtain B. 
 7) det(EA) = det(E)det(A) = det(A)det(B)

-How Row Operations Effect Determinants :

 Switch Rows         det(A) == -det(A)
 Elementary Adding   det(A) == det(A^T)
 Multiplying a row   det(A) == k*det(A)

- A is non singular iff det(A) does not equal 0 - Ax = 0 only has a non trivial solution iff det(A) = 0 - If A is non-singular then: det(A^-1) = 1/(det(A)) - If A and B are similar then: det(A) = det(B)

-Cofactor Expansion : This is another method to find a determinant. It is often used for matrices n >= 4.

 -general formula = (-1)^i+j * det(Mij)
  det(A) = ai1 Ai1 + ai2 Ai2 + ... + ain Ain   where a is the number in the matrix specified by the ith and jth entry and A is the determinant of the remaining matrix. If A = (4x4) and aij = a11, then Aij is going to be the determinant of the matrix with the first row(i) and the first column(j) crossed out. Essentaill one will multiply 4 numbers by 4 determinants to get the final answer of the total determinant. 
 - The sign is a checkerboard pattern of the matrix starting at with a (+) and at the location (a11).
 - The Aij is the co-factor of Axy.

-Inverse of a Matrix : One can find the inverse the exact same way as previously described.

 aj1*An1 if j does not equal k, then det = 0.

-Adjoint of a Matrix (adj(A)): It is the A^T of the cofactor matrix of A.

 1) Take the cofactor of A 
 2) Create a new matrix from the cofactors of A. (Lets call the new matrix B)
 3) Take the transpose of B to get the adjoint matrix (B^T)

-Cramers Rule : x*n = det(A') / det(A)

 det(A') is a new matrix where column b, the result gets place into x1, x2, ..., xn appropriately.

-Therom 3.12 : For A = (nxn) A(adjA) = (adj(A))*A = det(A)*In


Chapter 4


--Fquasnic 22:47, 9 December 2010 (UTC)


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