CS223 · Annotated SlidesChapter 1 — Vector Equations
Chapter 1 · Annotated slides

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.

Slide text Study notes Highlight: key words Tips, tricks & traps Seen again elsewhere How it can come up
Slide 1: Vector Equations
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    Slide 2: Vector Equations
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      Example

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        Slide 4: Geometric Descriptions of 2 R
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          Geometric Descriptions of 2 R

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            Slide 6: Example
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              Slide 7: Example
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                Slide 8: Vectors in 3 R
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                  Vectors in n R

                  • n ×1 colomn matrices with n entries n is the collection of all ordered n-tuples
                  • R
                  Vectors in n R
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                    Slide 10: Slide 10
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                      Slide 11: Linear Combinations
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                      1. y = c₁v₁ + … + cₚvₚ is a linear combination with weights c₁ … cₚ.
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                        Slide 13: Linear Combinations
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                        1. b is a linear combination of a₁, …, aₙ ⇔ the augmented matrix [a₁ … aₙ | b] is consistent.
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                        Written

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

                        Slide 14: Example
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                          Slide 15: Slide 15
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                            The following questions are equivalent:

                            • Is vector b in Span { v, , v }? ... 1 p
                            • Is there a solution for the equation
                            • Is there a solution for the augmented matrix
                            • And some other questions explained next .. ......
                            The following questions are equivalent:
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                              Slide 17: A Geometric Description of Span{v} and Span{u, v}
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                              1. Span{v} = a line through the origin; Span{u, v} (not parallel) = a plane through the origin.
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                              True / False

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

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                                Slide 19: Linear Combinations in Applications
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                                  Slide 20: Linear Combinations in Applications
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                                    The Matrix Equation Ax=b

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                                    1. The vector equation x₁a₁ + … + xₙaₙ = b, the matrix equation Ax = b and the augmented matrix all describe the same system.
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                                      Slide 23: Now, you know different ways to express a linear system
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                                        Slide 24: Slide 24
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