CS223 · Chapter 1 · Annotated SlidesVector Equations
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Vector Equations

Vectors in ℝ², ℝ³ and ℝⁿ, the parallelogram rule, linear combinations, writing a system as a vector equation, Span{v₁, …, vₚ} and its geometric meaning.

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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 25 slides
  1. 1 Vector Equations
  2. 2 Vector Equations
  3. 3 Example
  4. 4 Geometric Descriptions of 2 R
  5. 5 Geometric Descriptions of 2 R
  6. 6 Example
  7. 7 Example
  8. 8 Vectors in 3 R
  9. 9 Vectors in n R
  10. 10 is denoted by
  11. 11 Linear Combinations
  12. 12 Slide 12
  13. 13 Linear Combinations
  14. 14 Example
  15. 15 Slide 15
  16. 16 The following questions are equivalent:
  17. 17 A Geometric Description of Span{v} and Span{u, v}
  18. 18 This is also another question that
  19. 19 Linear Combinations in Applications
  20. 20 Linear Combinations in Applications
  21. 21 The Matrix Equation Ax=b
  22. 22 Matrix
  23. 23 Now, you know different ways to express a linear system
  24. 24 Slide 24
  25. 25 Slide 25
Slide 1 / 25

Vector Equations

Slide 1: Vector Equations
Slide 2 / 25

Vector Equations

Slide 2: Vector Equations
Slide 3 / 25

Example

Slide 3: Example
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Geometric Descriptions of 2 R

Slide 4: Geometric Descriptions of 2 R
Slide 5 / 25

Geometric Descriptions of 2 R

Slide 5: Geometric Descriptions of 2 R
Slide 6 / 25

Example

Slide 6: Example
Slide 7 / 25

Example

Slide 7: Example
Slide 8 / 25

Vectors in 3 R

Slide 8: Vectors in 3 R
Slide 9 / 25

Vectors in n R

Slide 9: Vectors in n R
Slide 10 / 25

is denoted by

Slide 10: is denoted by
Slide 11 / 25

Linear Combinations

Slide 11: Linear Combinations
Slide 12 / 25

Slide 12

Slide 12: Slide 12
Slide 13 / 25

Linear Combinations

Slide 13: Linear Combinations

Exam angle

PredictionQ1

How do you decide whether b is a linear combination of a₁ and a₂?

Compare with the model answer

Row reduce [a₁ a₂ | b]. If the system is consistent (no row [0 0 | nonzero]), b is a linear combination and the solution gives the weights; otherwise it is not.

Slide 14 / 25

Example

Slide 14: Example
Slide 15 / 25

Slide 15

Slide 15: Slide 15
Slide 16 / 25

The following questions are equivalent:

Slide 16: The following questions are equivalent:
Slide 17 / 25

A Geometric Description of Span{v} and Span{u, v}

Slide 17: A Geometric Description of Span{v} and Span{u, v}

Exam angle

PredictionQ1

Span{u, v} always contains the zero vector.

Slide 18 / 25

This is also another question that

Slide 18: This is also another question that
Slide 19 / 25

Linear Combinations in Applications

Slide 19: Linear Combinations in Applications
Slide 20 / 25

Linear Combinations in Applications

Slide 20: Linear Combinations in Applications
Slide 21 / 25

The Matrix Equation Ax=b

Slide 21: The Matrix Equation Ax=b

Notes

  1. The vector equation x₁a₁ + … + xₙaₙ = b, the matrix equation Ax = b and the augmented matrix all describe the same system.

Slide 22 / 25

Matrix

Slide 22: Matrix
Slide 23 / 25

Now, you know different ways to express a linear system

Slide 23: Now, you know different ways to express a linear system
Slide 24 / 25

Slide 24

Slide 24: Slide 24
Slide 25 / 25

Slide 25

Slide 25: Slide 25