Recipes of Arrows: Linear Combinations

Lesson 5 of 12 · 7 min · Free, no account needed

What you will learn

A linear combination of a and b is a recipe s a + t b: stretch each, then add tip to tail.
If a and b point along different lines, recipes reach every point of the plane.
If they lie on one line, every recipe stays on that line.

Your game has only two moves: a jump and a dash. Can you reach every spot on the map?
You may use each move any amount, even backwards.
Mix the jump and the dash to reach each star.
s says how much jump, t how much dash. The path is drawn tip to tail.
Now the dash is broken. Can you still reach the star?
Try a few recipes. Then fix it.
A recipe of arrows has a name.
s a + t b is a linear combination of a and b. Here 2a + 1b = (3, 3).
If a and b point along different lines, there is a recipe for every point.
If they lie on one line, every recipe stays on that line.
Whole plane or just one line?
Each card has two moves. Tap a card, then the bin.
Two arrows on one line are multiples of each other, so every mix is a multiple too.
Design your own two moves.
They must reach the whole map: not on one line.
Two arrows that are not multiples of each other reach the whole plane.
Game: flag hunt with your two moves.
Find a recipe that lands in each ring. Fewest lever moves wins.
AI stores words as arrows. Recipes of them carry meaning.
Where does king - man + woman land?
king - man + woman = (6, 5) - (5, 1) + (1, 1) = (2, 5), the point of queen.
What is the first part of 3 · (1, 2) + 2 · (2, -1)?
s a + t b by parts: 3 × 1 + 2 × 2 = 7 and 3 × 2 + 2 × (-1) = 4.
Two arrows on different lines reach everywhere.
Paint random recipes with your pair and with the broken pair.

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