Transformations in 2D

Linear Algebra (Matrices) · Axiom Academy

Every rotation, reflection, scaling, and shear of the plane is captured by one matrix — and applying it is just multiplication. 1. Rotation: Turning the Whole Plane To rotate the plane counterclockwise by angle , send to and to . Those become the two columns of the rotation matrix. Watch the unit square turn — the blue and orange arrows are exactly the columns, sweeping to their new positions. 2. Reflection: Flipping Across a Line A reflection mirrors the plane across a line through the origin. Across the x -axis, stays put while flips to (0,-1) — so the second column gets a minus sign. Watch the square fold across the axis; every y -coordinate changes sign. . Matrix , which swaps the coordinates. 3. Scaling & Shear: The Determinant Is the Area A diagonal scaling matrix stretches along the axes; a shear slants the square into a parallelogram. In the animation the unit square first stretches, then shears — and the shaded region's area is always . That is the whole meaning of the determinant: the factor by which the transformation scales area. , so the square's area triples (it becomes a rectangle). , so area is preserved — the square slants into a parallelogram of the same area. Worked example — rotate the point (1,1) by . At , and , so the rotation matrix is . Apply it to (1,1) : The point (1,1) lands at (-1,1) — a quarter-turn counterclockwise, exactly as expected.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.