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KQ1223 : MATEMATIK KEJURUTERAAN II (ALJABAR LINEAR), INI HAL YANG SANGAT…

- - - - solve the linear systems by reducing to reduced row echelon form
    - - If B is a square matrix satisfying BA = I, then B = A-1
      - If B is a square matrix satisfying AB = I, then B = A-1
    - - If A is an n x n matrix, then the following statements are equivalent
        
        The reduced row echelon form of A is In
        
        A is expressible as a product of elementary matrices
        
        Ax = 0 has only the trivial solution
        
        Ax = b is consistent for every n x 1 matrix b.
        
        Ax = b has exactly one solution for every n x 1 matrix b
        
        A is invertible
  - - - A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix
        
        Diagonal Matrices
  - - - Ax = 0 has only the trivial solution
      - The reduced row echelon form of A is In
      - Inversion Algorithm
        
        To find the inverse of an invertible matrix A, find a sequence of elementary row operations that reduces A to the identity and then perform that same sequence of operations on In to obtain A-1
        
        Invertible and noninvertibles matrices
      - A is expressible as a product of elementary matrices
  - - - If A is invertible and n is a nonnegative integer
        
        A^-1 is invertible and ( A^-1 )^-1 =A.
      - If A is an invertible matrix, then AT is also invertible
      - If A and B are invertible matrices with the same size, then AB is invertible
  - - - Triangular Matrix
      - Diagonal Matrix
      - Row Vector (Row Matrix)
      - Identity Matrix
      - Column Vector (Column Matrix)
      - Zero matrix (Null matrix)
      - Square Matrix
      - Symmetric Matrix
    - - Two matrices are defined to be equal if they have the same size and their
        corresponding entries are equal
      - Two matrices are defined to be equal if they have the same size and their
        corresponding entries are equal
      - If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A.
      - If A is m x r matrix and B is an r x n matrix, then the product AB is the m x n matrix whose entries are determined as follows: To find the entry in row i from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together, and then add up the resulting products
      - Partitioned Matrices
        
        A matrix can be subdivided or partitioned into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.
        
        Partitioned Matrices or Block Matrix Multiplication
      - Matrix Multiplication by Columns
        and by Row
        
        Partitioning has many uses, one of which is for finding particular rows or columns of a matrix product AB without computing the entire produc
  - - - If a linear system has infinitely many solutions, then a set of parametric equations from which all solutions can be obtained by assigning numerical values to the parameters is called a general solution of the system.
    - - rule 2
        
        if the are row that consist entirely zero, they are place at the bottom row
      - rule 3
        
        In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.
      - rule 4
        
        Each column that contains a leading 1 has zeros everywhere else in that column
      - rule 1
        
        if a a row does not contain entirely zero, the first no zero number in thee row is 1(called the leading 1) =
    - - Gauss Jordan Elimination & Reduced Row Echelon Form
    - - Free Variable Theorem for Homogeneous Systems
        
        If a homogeneous linear system has n unknowns, and if the reduced row echelon form of
        its augmented matrix has r nonzero rows, then the system has n – r free variables
    - - A homogeneous linear system with more unknowns than equations has infinitely many solutions
  - - - x+y=4 or 5x+6y=10
    - - 2x_1+3x_2+4x_3=10 or 3x_1+5x_1 x_2=2
    - - type of solution
        
        no solution
        
        one solution
        
        infinitely many solution
    - - coefficient matrix
      - augmented matrix
    - - interchange between two row
      - Add a constant times one
        row to another.
      - multiply a row through a non zero constant
- - - - Span {V1,V2,...,Vr}=Span{W1,W2,...,Wk}
  - - - The solution set of a homogeneous system Ax = 0 of m equations in n unknowns is a subspace of Rn.
    - - If W1, W2, …, Wr are subspaces of a vector space V, then the intersection of these subspaces is also a subspace of V.
    - - (a) If u and v are vectors in W, then u + v is in W.
        (b) If k is a scalar and u is a vector in W, then ku is in W.
- - - - DEFINITION
        
        If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the submatrix that remains after the ith row and jth column are deleted from A. The number (-1)i + j Mij is denoted by Cij and is called the cofactor of entry aij.
    - - ad - bc is not equal to zero
        
        its called determinant of matrix A, det(A) = ad - bc
      - inverse of A
    - - DEFINITIONS
        
        If A is an n x n matrix, then the number obtained by multiplying the entries in any row of column of A by the corresponding cofactors and adding the resulting products is called the determinant of A, and the sums themselves are called cofactor expansions of A.
  - - - Theorem 2.2.1:
        Let A be a square matrix. If A has a row of zeros or a
        column of zeros, then det(A) = 0
      - Theorem 2.2.2:
        Let A be a square matrix. Then, det(A) = det( A^T)
      - Theorem 2.2.3:
        Let A be an n x n matrix.
        1) If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, then det(B) = k det(A)
        2) If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = -det(A)
        3) If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det(B) = det(A)
      - Theorem 2.2.4:
        Let E be an n x n matrix.
        1) If E results from multiplying a row of In by a nonzero number k, then det(E) = k
        2) If E results from interchanging two rows of In, then det(E) = -1
        3) If E results from adding a multiple of one row of In to another, then det(E) = 1
  - - - A and B are n x n matrices and k is any scalar
        1) det(kA) = k^n det(A)
        2) det(A+B) is not equal det(A) + det(B)
        3) det(AB) = det(A)det(B)
      - Let A, B, and C be n x n matrices that differ only in a single row, say rth, and assume that the rth row of C can be obtained by adding corresponding entries in the rth rows of A and B. Then det(C) = det(A) + det(B)
        The same result holds for columns.
      - Let B is an n x n matrix and E is an n x n elementary matrix, then det(EB) = det(E)det(B)
      - A square matrix A is invertible if and only if det(A) is not equal 0
      - If A and B are square matrices of the same size, then det(AB) = det(A)det(B)
      - If A is invertible, then det(A^-1) = 1/det(A)
    - - If A is any n x n matrix and Cij is the cofactor of Aij , then (gambar)
        is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj A
      - Adjoint method; A^-1 = 1(adj A) / det(A)
    - - DEFINITION
        
        If Ax = b is a system of n linear equations in n unknowns such that det(A) is not equal 0,
        (gambar)
        where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix
        (gambar)
    - - If A is an n x n matrix, then the following statements are equivalent.
        1) A is invertible.
        2) Ax =0 has only the trivial solution.
        3) The reduced row echelon form of A is In.
        4) A can be expressed as a product of elementary matrices.
        5) Ax = b is consistent for every n x 1 matrix b.
        6) Ax = b has exactly one solution for every n x 1 matrix b.
        7) det(A) is not equal to 0
- - - - Products of the form P^ −1 AP in which A and P are n × n matrices and P is invertible will be our main topic of study in this section. There are various ways to think about such products, one of which is to view them as transformations of the form
        A > P^ −1 AP
        in which the matrix A is mapped into the matrix P^ −1 AP. These are called similarity transformations. Such transformations are important because they preserve many properties of the matrix A. For example, if we let B = P ^−1 AP, then A and B have the same determinant since
        det(B) = det(P^ −1 AP)=det(P^ −1 )det(A)det(B)
        In general, any property that is preserved by a similarity transformation is called a similarity invariant and is said to be invariant under similarity. Table 1 lists the most important similarity invariants.
    - - Determinant > A and P −1 AP have the same determinant.
      - Invertibility > A is invertible if and only if P −1 AP is invertible.
      - Rank > A and P −1 AP have the same rank.
      - Nullity > A and P −1 AP have the same nullity.
      - Trace > A and P −1 AP have the same trace.
      - Characteristic polynomial > A and P −1 AP have the same characteristic polynomial.
      - Eigenvalues > A and P −1 AP have the same eigenvalues.
      - Eigenspace dimension > If λ is an eigenvalue of A (and hence of P −1 AP) then the eigenspace of A corresponding to λ and the eigenspace of P −1 AP corresponding to λ have the same dimension.
    - - Step 3. P −1AP will be a diagonal matrix whose successive diagonal entries are the eigenvalues λ1, λ2, …, λn that correspond to the successive columns of P.
        
        Step 2. If you ascertained that the matrix is diagonalizable, then form the matrix whose column vectors are the n basis vectors you obtained in Step 1.
        
        Step 1. Determine first whether the matrix is actually diagonalizable by searching for n linearly independent eigenvectors. One way to do this is to find a basis for each eigenspace and count the total number of vectors obtained. If there is a total of n vectors, then the matrix is diagonalizable, and if the total is less than n, then it is not.
  - - - A dynamical system is a finite set of variables whose values change with time. The value of a variable at a point in time is called the state of the variable at that time, and the vector formed from these states is called the state vector (or state) of the dynamical system at that time. Our primary objective in this section is to analyze how the state vector of a dynamical system changes with time.
    - - Stated informally, the probability that an experiment or observation will have a certain outcome is the fraction of the time that the outcome would occur if the experiment could be repeated indefinitely under constant conditions—the greater the number of actual repetitions, the more accurately the probability describes the fraction of time that the outcome occurs.
    - - In a Markov chain with an initial state of x(0), the successive state vectors are
        x(1)=Px(0),x(2)=Px(1),x(3)=Px(2),x(4)=Px(3),.....
        For brevity, it is common to denote x(k) by xk, which allows us to write the successive state vectors more briefly as
        x1=Px0,x2=Px1,x3=Px2,x4=Px3,....
        Alternatively, these state vectors can be expressed in terms of the initial state vector x0 as
  - - - If A is an n × n matrix, then a nonzero vector x in Rn is called an eigenvector of A (or of the matrix operator TA) if Ax is a scalar multiple of x; that is,
        Ax = λx
        for some scalar λ. The scalar λ is called an eigenvalue of A (or of TA), and x is said to be an eigenvector corresponding to λ.
    - - If A is an n × n matrix, then λ is an eigenvalue of A if and only if it satisfies the equation det(λI - A) = 0
        This is called the characteristic equation of A.
      - If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the entries on the main diagonal of A.
      - If A is an n × n matrix, the following statements are equivalent.
        1) λ is an eigenvalue of A.
        2) λ is a solution of the characteristic equation det(λI − A) = 0.
        3) The system of equations (λI − A)x = 0 has nontrivial solutions.
        4) There is a nonzero vector x such that Ax = λx.
    - - A square matrix A is invertible if and only if λ = 0 is not an eigenvalue of A.
        Proof
        Assume that A is an n × n matrix and observe first that λ = 0 is a solution of the characteristic equation
        λn + c1 λ ^ n-1 + ...... + cn = 0
        if and only if the constant term cn is zero. Thus, it suffices to prove that A is invertible if and only if cn ≠ 0. But
        det( λI - A) - λn + c1 λ^n-1 + ..... + cn
        or, on setting λ = 0,
        det(-A) = cn or (-1)^n det(A) = cn
        It follows from the last equation that det(A) = 0 if and only if cn = 0, and this in turn implies that A is invertible if and only if cn ≠ 0.
    - - λ = 0 is not an eigenvalue of A.
        
        The orthogonal complement of the row space of A is {0}.
        
        The orthogonal complement of the null space of A is Rn.
        
        A has nullity 0.
        
        A has rank n.
        
        The row vectors of A form a basis for Rn.
        
        The column vectors of A form a basis for Rn.
        
        The row vectors of A span Rn.
        
        The column vectors of A span Rn.
        
        The row vectors of A are linearly independent.
        
        The column vectors of A are linearly independent.
        
        det(A) ≠ 0.
        
        Ax = b has exactly one solution for every n × 1 matrix b.
        
        Ax = b is consistent for every n × 1 matrix b.
        
        A is expressible as a product of elementary matrices.
        
        The reduced row echelon form of A is In.
        
        Ax = 0 has only the trivial solution.
        
        A is invertible.
- - - - LU‐decomposition (or LU‐factorization)
        
        LU‐decomposition (or LU‐factorization)
        
        A=LU
        
        L=L'D
        
        A=LDU
        
        PLU-decomposition or PLU-factorization
        
        A=PLU
        
        *P=Q^-1