$Matrix Transformations in 2D infographic - Rotation, Scaling, Reflection, and Shear$

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Matrix Transformations in 2D

Rotation, Scaling, Reflection, and Shear

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A 2D linear transformation is a function that maps every point $(x, y)$ in the plane to a new point $(x', y')$ in a way that preserves lines and the origin. Such transformations can always be represented by a $2 \times 2$ matrix multiplication: $[x', y']^\top = M \cdot [x, y]^\top$ . The columns of the matrix tell you where the standard basis vectors $(1, 0)$ and $(0, 1)$ land after the transformation - a key insight for visualizing what a matrix does.

Common 2D transformations include rotation (by angle θ), scaling (stretching or shrinking along axes), reflection (flipping over a line), and shear (tilting one direction while keeping the other fixed). Composing transformations means multiplying their matrices - and order matters because matrix multiplication is not commutative. Linear transformations form the geometric foundation of computer graphics, physics simulations, and machine learning (neural network layers are affine transformations: a linear transformation plus a translation).

Key Facts

Rotation by θ: [[cos θ, −sin θ], [sin θ, cos θ]] (counterclockwise)
Scaling: [[sx, 0], [0, sy]] (scales x by sx, y by sy)
Reflection over x-axis: [[1, 0], [0, −1]]; over y-axis: [[−1, 0], [0, 1]]
Shear (horizontal): [[1, k], [0, 1]] shifts x by k times y
Determinant = area scaling factor; $\det = 0$ means the transformation collapses to a lower dimension
Composition: applying $T_1$ then $T_2 = T_2 \cdot T_1$ (right-most matrix acts first)

Vocabulary

Linear transformation: A mapping between vector spaces that preserves vector addition and scalar multiplication; representable by a matrix.
Determinant: A scalar value of a square matrix that represents the scaling factor of the area (in 2D) under the transformation; zero determinant means the transformation is singular.
Basis vector: A vector in a set that, through linear combinations, can represent any vector in the space; the standard basis in 2D is (1,0) and (0,1).
Composition: Applying two transformations in sequence; mathematically represented by matrix multiplication (right-to-left order).
Eigenvector: A non-zero vector that is only scaled (not rotated) by a linear transformation; the scaling factor is the corresponding eigenvalue.

Common Mistakes to Avoid

Reversing matrix multiplication order when composing transformations. Applying rotation then scaling is NOT the same as scaling then rotating. Matrix multiplication is not commutative: AB ≠ BA in general.
Forgetting that linear transformations always fix the origin. Translations (moving the origin to a new point) are NOT linear transformations - they require homogeneous coordinates or affine notation.
Confusing the rotation angle direction. The standard rotation matrix [[cos θ, −sin θ], [sin θ, cos θ]] rotates counterclockwise. Clockwise rotation uses −θ (or equivalently swaps the signs of the sin terms).
Thinking a determinant of zero just means no rotation. Det = 0 means the matrix is singular: it collapses the plane onto a line or point, destroying information. The transformation cannot be undone (no inverse).

Practice Questions

1 Write the 2x2 matrix that rotates points 90° counterclockwise. Apply it to the point (3, 1) and verify geometrically.
2 A transformation first scales x by 2 and y by 3, then rotates 45°. Write the single matrix representing the composition (in correct order).
3 If a transformation matrix has determinant = 4, what happens to the area of a triangle that is transformed by it?

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