14-240/Classnotes for Monday September 22

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Polar coordinates:

  • r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta
  • r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta

The Fundamantal Theorem of Algebra: a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0 where a_i \in C and a_i != 0 has a soluion z \in C In particular, z^{2} - 1 = 0 has a solution.


  • Forces can multiple by a "scalar"(number).

No "multiplication" of forces.


Definition of Vector Space: A "Vector Space" over a field F is a set V with a special element O_v \in V and two binary operations:

  • + : V \times V -> V
  • \times : V \times V -> V

s.t.

  • VS_1 : \forall x, y \in V, x + y = y + x.
  • VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z.
  • VS_3 : \forall x \in V, x + 0 = x.
  • VS_4 : \forall x \in V, \exists y \in V, x + y = 0.
  • VS_5 : \forall x \in V, 1 \times x = x.
  • VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x.
  • VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay.
  • VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx.

Scanned Lecture Notes by AM

Scanned Lecture Notes by Boyang.wu

File:W31.pdf