Two pictures of the same puzzle — final
From The Geometry of Linear Equations · Gilbert Strang
Two rules at once are two paths. The solution is where they meet — if they meet.
Original lecture on MIT OpenCourseWareSeries final
Two pictures of the same puzzle — final
18.06 · The Geometry of Linear Equations · Age 11
Three sittings, one puzzle: two rules at once. Paths that meet, arrows that mix, and the box named A.
This series must show: system of two linear equations · row picture: lines in the plane · unique / none / infinitely many meetings · column picture: linear combination · mixing two vectors to reach b · matrix form Ax = b · dependent columns / singular case
Diagnostic
1. Two rules at once. In the row picture, a solution is…
2. Two paths that never meet means the system has…
3. In the column picture, a solution (x, y) is…
4. Taking 3 of the arrow (2, 1) gives…
5. Ax = b is…
6. If the two columns of A point the same way, some targets are unreachable. That box is…
Show your work
7. Two paths: x + y = 4 and x − y = 2. Find the meeting. Then say the same answer as a mix of the columns of A = [[1, 1], [1, −1]] to reach b = (4, 2).
Hint: Add the two rules. Then check the mix.
In your own words
Same puzzle, two pictures. In one sentence each, say what a solution looks like as paths, and as arrows.
A situation you have not seen
8. A kitchen has two floor tapes that lie on top of each other (same line). You want to stand on a point that is not on that line. How many ways can you satisfy both tapes at once?