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Transformations: translation, rotation, reflection, dilation

Rigid motions and scalings, and the matrices that perform them.

Translations slide, rotations turn, reflections flip, dilations scale. The first three preserve distances (isometries); dilation preserves shape. Picture it: the 2×2 matrix [[0,−1],[1,0]] turns every vector a quarter turn — the 3D view shows the unit square rotating. Think it: transformations compose, invert and form a group, which is where abstract algebra begins.

Ohatra: inverse of [[0,-1],[1,0]]

Linear-Algebra · Matrix-Inverse

Inverse of [[0,-1],[1,0]]

\left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]

Dingana amin'ny dingana

  1. \det A = 1

    A matrix is invertible only when its determinant is non-zero.

  2. A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{1}\left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

    For 2×2: swap the diagonal, negate the off-diagonal, divide by the determinant.

  3. A^{-1} = \left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

Asehoy ny valinteny
A^{-1} = \left[\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right]

Symbols used here

\det A,\ |A|
determinant
Scaling factor of area/volume under A; zero means singular.
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
matrix
A rectangular array of numbers; a linear map.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\theta
theta
The usual name for an angle.
^\circ
degrees
1/360 of a full turn. 180° = π radians.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\angle ABC,\ \triangle ABC
angle, triangle
The angle at B between BA and BC; the triangle with those vertices.
\parallel,\ \perp,\ \cong,\ \sim
parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Transformations: translation, rotation, reflection, dilation

  1. Identify the transformation type from what is preserved: lengths (isometry) or only angles (similarity).
  2. Write it as a matrix acting on column vectors, plus a translation vector if there is one.
  3. Apply it to each vertex; compose by multiplying matrices in the order the moves happen (right to left).

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

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Mbola maro ao Geometry

TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryEuclid's axioms and the structure of proofCongruent and similar trianglesCircle theoremsSolid geometry: prisms, pyramids, spheres