ChristopheTUD
commented
1 month ago Suggestion for introductory text above the applet:
The following sets $\mathcal{A}$ (indicated in yellow) are no subspaces of $\mathbb{R}^2$, since they do not satisfy ii. and/or iii. from Definition \ref{}.
Do you see why? First pick a set $\mathcal{A}$ from the drop-down menu. Then, either choose vectors $\vec{u}$ and $\vec{v}$ in $\mathcal{A}$ such that $\vec{u+v}$ is not in $\mathcal{A}$ (left pane),
or
choose a vector $\vec{w}$ in $\mathcal{A}$ and a scalar $c$ such that $c\vec{w}$ is not in $\mathcal{A}$ (right pane).
Remarks:
- I just notice Fokko initially suggested the line x+y=1, so also a line not through the origin. That would make all cases non-subspaces, which is perfect.
- I called the vector on the right $\vec{w}$ rather than $\vec{u}$, to avoid confusion between the two panes.
- Maybe instead of the cones we could use the union of first and third quadrant? Then you first have the case of the first quadrant (which you already have), where the left pane does not give a counterexample, but the right one does (by choosing c to be negative). And then you go to the (first quadrant union third quadrant), where this problem is solved, but where now the left pane gives a counterexample.
Maybe we can discuss the third remark next week.
Link to paragraph
https://dbalague.pages.ewi.tudelft.nl/openlabook/Chapter4/Subspaces_of_Rn.html
What type of applet is this?
Interactive applet 2D
Link to applet
No response
What state is the static image?
No, Dennis has not created a static image yet
Static image
No response
Requirements
See below
What elements are interactive?
There is a way to select a set from a list (list below) and then a rule ("sum" or "multiple").
Then if rule "sum" is chosen the student can drag & drop two vectors v and w. Shown is v, w and sum v+w. If v, w in set and v+w not in set you get the message "Good counterexample showing it is not a subspace"; if v and/or w are not in the set "Please choose v and w in the set" and if v+w is in the set "This example does not prove whether this is a subspace or not".
If the rule multiple is chosen, the student can drag & drop one vector v, and has a slider for the multiple c (from -5 to 5 I guess). Shown are v and cv. If v is in the set and cv is not you get the message "Good counterexample showing it is not a subspace." If v is not in the set you get the message "Please choose v in the set". If cv is in the set you get the message "This example does not prove whether this is a subspace or not".
List of sets to try: Affine Line $x+y=1$; Disc $x^2+y^2\leq 1$ two axes $xy=0$
First quadrant $x,y \geq 0$ Two-sided cone $0\leq x\leq y$ union $y\leq x\leq 0$.
Make sure the vectors click on the set in the case of lines.
Does this issue relate or depend on other issues?
(Not checked with colleagues yet)
Steps to completion: