The Zero Route
Jewelhouse Observatory log
Find everything the map cannot see
In the Jewelhouse Observatory, Vivian compares two proposed signal maps for a dormant Translation Frame. Each map takes a displacement vector in and returns a two-number signal. Both maps are linear transformations.
Map A
Map B
Your route
From a system to a theorem
- solve each kernel
- draw what each map erases
- build a collision
- prove collision and fiber rules
- prove Vivian’s conjecture
Turn the page for the kernel record. Both map rules are repeated there for reference.
Page 2 · Kernel record
What does each map erase?
Show the system, the complete kernel, and its geometry.
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For each map, solve . Record the complete kernel in set notation, using a real parameter if needed. Name its geometry: the origin or a line through it.
Map Homogeneous equations and work in set notation Geometry A B -
Draw each kernel. If it is only the origin, mark the origin with a solid dot. If it is a line, draw the entire line across the grid. When the kernel contains a nonzero vector, mark one example and verify by substitution that the map sends it to zero.
Substitution check
Show the verification for your marked vector here.
Part 3 · Build a collision
Two inputs, one signal
For Map B, find two different inputs and that produce the same signal. Write the complete equality .
(, )
(, )
(, )
Page 3 · The collision rule
Move every collision to zero
“A list of successful tests is not a proof.”
One-to-one
For a map , is one-to-one (injective) when different inputs always give different outputs. Equivalently,
Fiber
For and , the fiber over is the set of all inputs that produce :
Here denotes the preimage of ; need not have an inverse function. If and , then is the translate of by .
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Let be linear, and let . Prove both directions of if and only if is in . Now suppose the fiber over is nonempty. Choose with , and prove .
You may use: .
Start with
Start with
Fiber proof.
Show both directions: if , then ; if with , then .
On page 4, use this collision rule to prove Vivian’s conjecture.
Page 4 · Vivian’s conjecture
Why zero is enough
Use the collision rule below as your theorem-building tool.
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Using the collision rule, prove both directions of Vivian’s conjecture. Then identify which proposed map is one-to-one and state precisely what the kernel records about the input information a linear transformation erases.
Continue on separate paper if needed.
One-to-one
one-to-one
Which proposed map is one-to-one?
In one precise sentence, what does the kernel record?