← ML Foundations

Learn / ML Foundations

ML Foundations: Linear Transformations and Rotation Matrices

Foundational quiz on linear maps, geometric effects of matrices, 2D rotations, and transformation properties.

Learning path

0%

0 of 4 sections marked complete · about 25 minutes

Learning objectives

What you will be able to explain

  • What makes a transformation linear?
  • Examples of transformations: true or false
  • Which matrix represents a 2D rotation by angle θ\theta?
  • Rotate a vector by 90∘90^\circ
  • Rotate a vector by 45∘45^\circ
  • What is the inverse of a rotation matrix R(θ)R(\theta)?

Section 01

What makes a transformation linear? to Which matrix represents a 2D rotation by angle θ\theta?

01

What makes a transformation linear?

Linearity means T(x+y)=T(x)+T(y)T(x + y) = T(x) + T(y) and T(cx)=cT(x)T(cx) = cT(x). This is why linear transformations can be represented by matrices.

Guided checkpoint

Choose the defining property.

02

Examples of transformations: true or false

Rotations and scalings preserve the algebraic structure required for linearity. A nonzero translation breaks T(0)=0T(0) = 0, and squaring coordinates breaks additivity.

Guided checkpoint

Mark whether each transformation is linear.

03

Which matrix represents a 2D rotation by angle θ\theta?

The standard 2D rotation matrix is R(θ)=[cos⁡θ−sin⁡θsin⁡θcos⁡θ]R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}. It rotates vectors around the origin without changing their length.

Guided checkpoint

Choose the standard rotation matrix.

Section 02

Rotate a vector by 90∘90^\circ to What is the inverse of a rotation matrix R(θ)R(\theta)?

01

Rotate a vector by 90∘90^\circ

A 90∘90^\circ counterclockwise rotation maps the positive x-axis to the positive y-axis, so (1,0)(1, 0) becomes (0,1)(0, 1).

Guided checkpoint

Rotate (1,0)(1, 0) counterclockwise by 90∘90^\circ.

02

Rotate a vector by 45∘45^\circ

Applying R(45∘)R(45^\circ) to (1,1)(1, 1) gives (cos⁡45∘−sin⁡45∘,sin⁡45∘+cos⁡45∘)=(0,2)(\cos 45^\circ - \sin 45^\circ, \sin 45^\circ + \cos 45^\circ) = (0, \sqrt{2}).

Guided checkpoint

Rotate (1,1)(1, 1) counterclockwise by 45∘45^\circ.

03

What is the inverse of a rotation matrix R(θ)R(\theta)?

Rotation matrices are orthogonal, so R(θ)−1=R(θ)⊤=R(−θ)R(\theta)^{-1} = R(\theta)^\top = R(-\theta). Geometrically, the inverse rotation undoes the original angle.

Guided checkpoint

Choose the correct answer.

Section 03

How do 2D rotations compose? to Match each transformation to its effect

01

How do 2D rotations compose?

Successive rotations add their angles. This is one reason rotation matrices are such a clean algebraic model of geometric transformations.

Guided checkpoint

Choose the correct identity.

02

Properties of rotation matrices

Rotations are rigid motions around the origin: they preserve lengths, inner products, and angles. That is exactly why they are represented by orthogonal matrices with determinant 1.

Guided checkpoint

Mark each statement as true or false.

03

Match each transformation to its effect

These are standard geometric effects of linear transformations. Thinking in terms of geometry helps make matrix operations intuitive in ML.

Guided checkpoint

Match the transformation type to the best description.

Section 04

Rotate by 180∘180^\circ

01

Rotate by 180∘180^\circ

A 180∘180^\circ rotation sends every vector to its negative, so (2,−3)(2, -3) becomes (−2,3)(-2, 3).

Guided checkpoint

Rotate (2,−3)(2, -3) counterclockwise by 180∘180^\circ.

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

Start the quiz →Back to course