14-240/Classnotes for Monday September 22

Polar coordinates:

• [math]\displaystyle{ r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta }[/math]
• [math]\displaystyle{ r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta }[/math]

The Fundamantal Theorem of Algebra: [math]\displaystyle{ \a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0 }[/math] where [math]\displaystyle{ \a_i \in C and \a_i != 0 }[/math] has a soluion [math]\displaystyle{ z \in C }[/math] In particular, [math]\displaystyle{ z^{2} - 1 = 0 }[/math] 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 [math]\displaystyle{ \O_v \in V }[/math] and two binary operations:

• [math]\displaystyle{ + : V \times V -\gt V }[/math]
• [math]\displaystyle{ \times : V \times V -\gt V }[/math]

s.t.

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