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A linear map is geometry with discipline. It can stretch,

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shear, or rotate space, but two promises never break: straight

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lines stay straight, and the origin never moves.

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Start with the standard grid and the two basis

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arrows, e one and e two. They are the

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coordinate rulers: every vector is built from these directions.

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Now add the unit square. Right now its area is

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one, and its edges align with the basis. This square

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will show us exactly what the transformation does.

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Apply the matrix A. The first basis arrow lands at the first column,

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and the second arrow lands at the second column. Columns are the map.

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Watch the full grid move at once. Cells tilt, squares

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become parallelograms, and distances change unevenly. But every row and

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column remains a family of lines - classic shear behavior.

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Notice what survives: this diagonal was a line before, and

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it is still a line after. And the origin, where

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both basis arrows start, is still pinned in place.

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The unit square became a larger parallelogram. Its area scale is

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the determinant. Here det A is one point four, so every

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tiny area patch gets multiplied by one point four.

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Now lift the idea to three dimensions. A unit cube is

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just three basis directions combined. The same linear rules still apply

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in three D: planes stay planes, and the origin stays fixed.

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When this transformed cube appears in graphics, robotics, or

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data embeddings, it is the same story: new basis

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vectors first, then every point follows by linear combination.

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That is a linear map: choose where the basis goes,

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and everything else is forced. Structure first, motion second.