14-240/Classnotes for Monday September 22

Polar coordinates:

• $r\times e^{i\theta }=r\times cos\theta +i\times rsin\theta$
• Failed to parse (unknown function "\r"): {\displaystyle \r_1 \times e^{i\\theta_2} = \r_1 \times (cos\theta + sin\theta}

The Fundamantal Theorem of Algebra: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0} where Failed to parse (unknown function "\a"): {\displaystyle \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 $\emptyset _{v}\in V$ and two binary operations:

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

s.t.

• Failed to parse (unknown function "\VS"): {\displaystyle \VS_1 : \forall x, y \in V, x + y = y + x} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_3 : \forall x \in V, x + 0 = x} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_4 : \forall x \in V, \exists y \in V, x + y = 0} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_5 : \forall x \in V, 1 \times x = x} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x} .
• Failed to parse (unknown function "\VS"): {\displaystyle \VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay} .
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx} .