14-240/Classnotes for Monday September 22
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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 }[/math]and[math]\displaystyle{ 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].
Scanned Lecture Notes by AM
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