1st major update src/3_vector_spaces.md

src/1_complex_numbers.md

@@ -182,7 +182,7 @@ Furthermore, we can define the sine and cosine in terms of complex exponentials:
     $$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
 
 Most operations on complex numbers become easier when complex numbers are converted to their *polar form* using the complex exponential.
-Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by sustituting the real variable $x$ with the complex variable $z$ in its polar form:
+Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by substituting the real variable $x$ with the complex variable $z$ in its polar form:
 !!! info "Examples of some complex functions stated using polar form"
     $$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
     $$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
@@ -219,7 +219,7 @@ We can then regard the complex ${\rm i}$ as another constant, and use our usual
 
 ## 1.4. Bonus: the complex exponential function and trigonometry
 
-Let us show some tricks in the folloiwing examples where the simple properties of the exponential
+Let us show some tricks in the following examples where the simple properties of the exponential
 function help in re-deriving trigonometric identities.
 
 !!! example "Properties of the complex exponential function I"
@@ -300,7 +300,7 @@ function help in re-deriving trigonometric identities.
     (b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
     for real $a$ and $b$.
 
-5.  [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geomtric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
+5.  [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geometric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
 
 6.  [:grinning:] Compute (a) 
     $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$