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
- Lines in two-dimensions
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