CS223 · Chapter 1 · Annotated SlidesLinear Transformations
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Overview

Linear Transformations

Matrix transformations x ↦ Ax, domain, codomain and range, linearity and superposition, shear, rotation, reflection and projection examples, the standard matrix A = [T(e₁) … T(eₙ)], and one-to-one and onto transformations.

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Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.

All 33 slides
  1. 1 Introduction to Linear Transformation
  2. 2 Introduction to Linear Transformation
  3. 3 Slide 3
  4. 4 Slide 4
  5. 5 Slide 5
  6. 6 Slide 6
  7. 7 Slide 7
  8. 8 We need to find whether has a unique solution
  9. 9 Slide 9
  10. 10 Slide 10
  11. 11 Example: shear transformation
  12. 12 Linear Transformations
  13. 13 Superposition
  14. 14 Why Linear Transformation is useful?
  15. 15 Why Linear Transformation is useful?
  16. 16 Another solution:
  17. 17 Slide 17
  18. 18 Slide 18
  19. 19 The Matrix of a Linear Transformation
  20. 20 The Matrix of a Linear Transformation
  21. 21 Slide 21
  22. 22 Slide 22
  23. 23 1. This transformation rotates a vector input
  24. 24 Geometric Linear Transformations of
  25. 25 Existence and Uniqueness Questions
  26. 26 b
  27. 27 b
  28. 28 Existence and Uniqueness Questions
  29. 29 A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides
  30. 30 A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides 1. This transformation description:
  31. 31 Slide 31
  32. 32 Slide 32
  33. 33 Slide 33
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Introduction to Linear Transformation

Slide 1: Introduction to Linear Transformation
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Introduction to Linear Transformation

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We need to find whether has a unique solution

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Example: shear transformation

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Linear Transformations

Slide 12: Linear Transformations

Notes

  1. T is linear ⇔ T(u + v) = T(u) + T(v) and T(cu) = cT(u). Consequently T(0) = 0.

  2. Memory trick Quick test: if T(0) ≠ 0, T is NOT linear.

Exam angle

PredictionQ1

Which map is NOT linear?

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Superposition

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Why Linear Transformation is useful?

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Why Linear Transformation is useful?

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Another solution:

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The Matrix of a Linear Transformation

Slide 19: The Matrix of a Linear Transformation

Exam angle

PredictionQ1

Find the standard matrix of the rotation of ℝ² by 90° counter-clockwise.

Compare with the model answer

T(e₁) = (0, 1), T(e₂) = (−1, 0), so A = [[0, −1], [1, 0]].

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The Matrix of a Linear Transformation

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1. This transformation rotates a vector input

Slide 23: 1. This transformation rotates a vector input
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Geometric Linear Transformations of

Slide 24: Geometric Linear Transformations of

Notes

  1. Geometric maps: reflections, contractions/expansions, shears, projections (see the tables).

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Existence and Uniqueness Questions

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Notes

  1. Onto ℝᵐ ⇔ columns span ℝᵐ ⇔ pivot in every row. One-to-one ⇔ T(x) = 0 has only the trivial solution ⇔ columns independent ⇔ pivot in every column.

Exam angle

Past exam · Major 1 (251)

Related question: this topic is examined

PredictionQ1

T: ℝ³ → ℝ² can never be:

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b

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b

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Existence and Uniqueness Questions

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A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides

Slide 29: A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides
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A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides 1. This transformation description:

Slide 30: A yet. So, we find first A We don t have the standard matrix ’ Consult the following slides 1. This transformation description:
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