Computational

• Equation


  An expression of the shape A = B, where A and B are expressions containing one or several variables called unknowns

  
• Algebraic equation


  An equation in which both sides are polynomials
• System of polynomial equations


  A set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k
• Linear equation


  An algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable
• Linear equation


  • General (or standard) form: Ax + By = C, where A and B are not both equal to zero
  • Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept
  • Point-slope form: y - y1 = m(x - x1), where m is the slope and (x1,y1) is any point on the line
  • Two-point form: (y2 - y1)/(x2 - x1) = (y - y1)/(x - x1), where (x1, y1) and (x2, y2) are two points on the line
  • Intercept form: x/a + y/b = 1, where a and b must be nonzero
  • Matrix form: Ax = b, where A is a matrix and x and b are vectors
• Quadratic equation


  A univariate polynomial equation of the second degree, in the form ax^2 + bx + c = 0, where a, b, and c are constants with a ≠ 0

  
• Methods to solve quadratic equations

  1. Quadratic formula

  2. Factoring

  3. Completing the square

  4. Graphing

  
• The solution of a quadratic equation is the value of x when you set the equation equal to zero

  
• The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / (2a)

  
• Factoring a quadratic equation involves finding two factors of the constant term c that add up to the coefficient b

  
• Solving a quadratic equation by factoring

  Step 1) Create a factor chart to find factor pairs of c

  Step 2) Out of all of the factor pairs from step 1, look for the pair (if it exists) that add up to b

  Step 3) Insert the pair you found in step 2 into two binomials

  Step 4) Solve each binomial for zero to get the solutions of the quadratic equation

  
• Perfect square trinomial


  A polynomial that you get by squaring a binomial
• Completing the square


  A method for solving quadratic equations

  
• In a quadratic equation x² + bx + c that is a square of a binomial, c is always the square of ½(b)

  
• Solving a system of equations

  Step 1) Graphing

  Step 2) Substitution

  Step 3) Elimination or addition method

  Step 4) Matrices
• Simultaneous equations


  Conditions where two or more unknown variables are related to each other through an equal number of equations
• Matrix


  • - An array of numbers arranged into a fixed number of rows and columns
  • The numbers of rows and columns are called the dimensions of the matrix
  • Types of matrices include: square matrix, diagonal matrix, row matrix, column matrix, 0-matrix, identity matrix, scalar matrix
• Matrix operations


  • Addition: Add the numbers in the matching positions
  • Subtraction: Subtract the numbers in the matching positions
  • Multiplication: Multiply the rows of the first matrix with the columns of the second matrix
• Matrix


  • 3 rows and 5 columns
  • 3 rows and 4 columns
• Sum of two matrices A and B


  C = A + B, defined by adding entries with the same indices
• Subtracting two matrices


  • Subtract the numbers in the matching positions
• Subtracting is actually defined as the addition of a negative matrix: A + (-B)
• Scalar multiplication


  Multiplying a matrix by a single number
• Scalar multiplication


  • 2×4=8
  • 2×0=0
  • 2×1=2
  • 2×-9=-18
• Calculating sales value for Monday

  • Beef pie value + Chicken pie value + Vegetable pie value

  $3×13 + $4×8 + $2×6 = $83
• Dot product


  Matching the price to how many sold, multiplying each, then summing the result
• Sales for Monday, Tuesday, Wednesday, Thursday in matrix form
• Rows and columns


  To show how many rows and columns a matrix has, we write rows×columns
• Matrix multiplication


  • The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix

  The result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix
• Matrix multiplication is not commutative: AB ≠ BA
• Identity matrix


  Square matrix with 1s on the diagonal and 0s everywhere else
• When multiplying by the identity matrix, the original is unchanged: A × I = A, I × A = A
• Determinant of a matrix


  A special number that can be calculated from a square matrix
• Determinant symbol


  |A| means the determinant of the matrix A
• Determinant of a 2x2 matrix


  |A| = ad - bc
• Cramer's rule


  Used to solve systems with matrices
• Inverse of a matrix

  The reciprocal of a matrix, denoted A^-1
• When multiplying a matrix by its inverse, the result is the identity matrix: A × A^-1 = I
• Conditions for a matrix to have an inverse


  • The matrix must be square (same number of rows and columns)

  The determinant cannot be zero
• A matrix with determinant zero has no inverse and is called singular