Added Nikita notes

Advanced/Linear Maps/main.tex

@@ -15,6 +15,18 @@
 	fit
 }
 
+
+% Let's give clarifications about the meaning of Z and R when we use them in the first problems.
+
+% Definitely define $R^n$ before using. Optionally you may add a problem "convince yourself that $R^2$ is a plane and $R^3$ is a 3-d space".
+
+% Maybe we can add an example of a linear transformation from R^2 to R^2? Rotation? Scaling of y-axis?
+
+% Slow down, understand linear transformations fully.
+
+
+
+
 %\usepackage{lua-visual-debug}
 \renewcommand{\arraystretch}{1.2}
 \begin{document}

Advanced/Linear Maps/parts/2 linear.tex

@@ -24,20 +24,15 @@ Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear.
 Is $f(x) = mx + b$ a linear map on $\mathbb{R}$?
 
 \vfill
+
 \problem{}
-In general, what does a linear map in $\mathbb{R}^n$ look like?
+In general, what does a linear map $\mathbb{R} \to \mathbb{R}$ look like?
 
 \vfill
 
 \problem{}
-Is $\text{median}(v): \mathbb{R}^n \to \mathbb{R}$ a linear map on $\mathbb{R}^n$?
+Is the map ${median}(v): \mathbb{R}^3 \to \mathbb{R}$ linear? \\
-
+\hint{$median([3, 5, 4]) = 4$, but you already knew that.}
-
-\vfill
-
-\problem{}
-Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\
-\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $n$.}
 
 \vfill
 \pagebreak

Advanced/Linear Maps/parts/3 matrices.tex

@@ -181,18 +181,29 @@ Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be wr
 
 \vfill
 
-
-\problem{}
-Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}p$. \\
-Find a matrix that corresponds to $D$. \\
-\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
-
-
 \vfill
 \pagebreak
 
 \problem{}
-Show that the set of all linear maps is a vector space.
+Show that $\mathbb{P}^n$ is a vector space.
+\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $ \leq n$.}
+
+\vfill
+
+\problem{}
+Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\
+
+\vfill
+
+\problem{}
+Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}(p)$. \\
+Find a matrix that corresponds to $D$. \\
+\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
+
+\vfill
+
+\problem{}
+Show that the set of all linear maps $\mathbb{R}^n \to \mathbb{R}^m$ is a vector space.
 
 \vfill
 \pagebreak