Review of Numerical Methods

Basics:

  •  Vectors:
    • plus, minus, scalar multiplication
    • magnitude and direction
    • inner product and angle
      • Cauchy-Schwarz inequality
    • parallel and orthogonal
    • projection
    • cross product
      • orthogonal to both the two input vectors
      • right-hand rule
      • magnitude is the area size of the parallelogram, or twice of the area size of the triangle
  • Linear equations
    • Lines in two-dimensions
      • Parameterization: define a line: basepoint, direction vector
      • Direction vector for Ax+By=k is [B, -A]. A normal vector is [A, B].
      • Two lines are parallel if their normal vectors are parallel vectors.
      • If two lines are not parallel, then they have a unique intersection.
      • If two lines are parallel, they may not have intersection at all or the same line with infinitely many intersections.
      • Two parallel lines are equal, <=> the vector connecting one point on each line is orthogonal to the lines’ normal vectors.
      • If two non-parallel lines Ax+By=k1, Cx+Dy=k2; then A and C have one zero at most, AD-BC!=0.
        • The intersection is x=(Dk1-Bk2)/(AD-BC); y=(-Ck1+Ak2)/(AD-BC).
      • Use normal vectors is better for high dimensions.
    • Planes in three dimensions
      • Ax+By+Cz=k,
      • Normal vector: [A,B,C]
      • If two planes are equal <=> the vector connecting one point on each plane is orthogonal to the planes’ normal vectors.
      • Given Ax+By+Cz=k1, Dx+Ey+Fz=k2, possible solutions sets are:
        • a line with direction vector [A,B,C] x [D,E,F], if planes are not parallel;
        • no solutions, if planes are parallel but not equal;
        • a plane, if the planes are the same.
      • More planes could intersect in a single point.
      • We need at least two lines in two variables to obtain a unique intersection; We need at least three planes in three variables to obtain a unique intersection.
      • Rules for manipulating equations
        • Should preserve the solution, should be reversible
        • swap order of equations
        • multiply an equation by a nonzero number
        • Add a multiple of an equation to another
      • A system is inconsistent <=> we find 0=k for k nonzero during Gaussian elimination.
      • It’s not enough to count # of equations (usually) or look for 0=0 to determine if infinitely many solutions.
      • A consistent system has a unique solution <=> each variable is a pivot variable.
      • #Free variables = dimension of solution set
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