# 14-240/Classnotes for Monday September 22

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.

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