diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex index 914fb41..be3e488 100755 --- a/Advanced/Linear Maps/main.tex +++ b/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} diff --git a/Advanced/Linear Maps/parts/2 linear.tex b/Advanced/Linear Maps/parts/2 linear.tex index 0720333..8c2be8d 100644 --- a/Advanced/Linear Maps/parts/2 linear.tex +++ b/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 $\mathbb{R}^n$? - - -\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 $n$.} +Is the map ${median}(v): \mathbb{R}^3 \to \mathbb{R}$ linear? \\ +\hint{$median([3, 5, 4]) = 4$, but you already knew that.} \vfill \pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex index 6dc2ce3..d3b2bea 100644 --- a/Advanced/Linear Maps/parts/3 matrices.tex +++ b/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 \ No newline at end of file