Once we accept that linearity in matrices requires:

• Positional integrity: each entry must remain anchored to its row and column
• Proportional scaling: all entries must scale uniformly, preserving internal ratios

Then several derived conditions naturally emerge:

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📐 Shape Preservation

Matrices must be of the same shape to be added or scaled
→ Ensures positional alignment across all entries

$$ A, B \in \mathbb{R}^{m \times n} \quad\Rightarrow\quad A + B \text{ is defined} $$

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➕ Closure Under Addition

The sum of two matrices in the same space remains in that space
→ Ensures that linear combinations stay within the same abstraction

$$ A, B \in V \quad\Rightarrow\quad A + B \in V $$

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🔁 Homogeneity

Scaling a matrix by a scalar scales the output uniformly
→ Preserves proportional relationships

$$ f(kA) = kf(A) $$

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➕ Additivity

The transformation of a sum equals the sum of transformations
→ Preserves superposition logic

$$ f(A + B) = f(A) + f(B) $$

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