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doc: add linear map definition to tactic docs #17

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18 changes: 14 additions & 4 deletions Game/Levels/LinearMapsWorld/Level01_backup.lean
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Expand Up @@ -26,13 +26,13 @@ Given vector spaces $V$ and $W$ over a field $K$, a function $T : V \\to W$ is c
Linear maps are the structure-preserving functions of linear algebra. They respect the vector space structure, making them the natural morphisms between vector spaces.

### Your Goal
Prove that our definition captures exactly these two fundamental properties.
Prove that our definition captures exactly these two fundamental properties.

In Lean, we define `is_linear_map_v K V W T` (see Definitions panel) to formalize exactly what it means for a function T to be linear.
"

open VectorSpace
variable (K V W : Type) [Field K] [AddCommGroup V] [AddCommGroup W]
variable (K V W : Type) [Field K] [AddCommGroup V] [AddCommGroup W]
variable [DecidableEq V] [DecidableEq W] [VectorSpace K V] [VectorSpace K W]

/--
Expand All @@ -45,11 +45,21 @@ This follows Axler's Definition 3.1: A function T : V → W is called a linear m
T(u + v) = Tu + Tv and T(av) = aTv for all u, v ∈ V and all a ∈ F.
-/
def is_linear_map_v (T : V → W) : Prop :=
(∀ u v : V, T (u + v) = T u + T v) ∧
(∀ u v : V, T (u + v) = T u + T v) ∧
(∀ a : K, ∀ v : V, T (a • v) = a • T v)

/--
`is_linear_map_v K V W T` means T preserves addition and scalar multiplication.

# Mathematically
• Additivity: $T(u + v) = T(u) + T(v)$ for all $u, v \\in V$

• Homogeneity: $T(a \\cdot v) = a \\cdot T(v)$ for all $a \\in K$ and $v \\in V$

# Lean
```
((∀ (u v : V), T (u + v) = T u + T v) ∧ ∀ (a : K) (v : V), T (a • v) = a • T v)
```
-/
DefinitionDoc is_linear_map_v as "is_linear_map_v"

Expand All @@ -66,7 +76,7 @@ TheoremTab "Linear Maps"
The definition of a linear map is exactly additivity and homogeneity.
-/
Statement linear_map_def (T : V → W) :
is_linear_map_v K V W T ↔
is_linear_map_v K V W T ↔
(∀ u v : V, T (u + v) = T u + T v) ∧ (∀ a : K, ∀ v : V, T (a • v) = a • T v) := by
Hint "Try unfold is_linear_map_v to see the definition directly."
Hint (hidden := true) "Try `unfold is_linear_map_v`"
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