CS223 · Annotated SlidesChapter 1 — The Matrix Equation Ax = b and Solution Sets
Chapter 1 · Annotated slides

The Matrix Equation Ax = b and Solution Sets

The product Ax as a linear combination of A’s columns, the row-vector rule, properties of Ax, homogeneous systems and nontrivial solutions, and solution sets in parametric vector form (p + vₕ).

Slide text Study notes Highlight: key words Tips, tricks & traps Seen again elsewhere How it can come up
Slide 1: Slide 1
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    Slide 2: A Computation of Matrix-Vector Product x
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    1. Ax = x₁a₁ + … + xₙaₙ (columns weighted by x). Defined only if A has as many columns as x has entries.
    Slide 3: A Computation of Matrix-Vector Product x
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      Slide 4: Example
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        Slide 5: Ax Properties of the Matrix-Vector Product
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        1. A(u + v) = Au + Av and A(cu) = c(Au).

        Solution Set of Linear Systems

        Homogeneous Linear Systems: System of linear equations that can be written in the form A homogeneous Linear System always has at least one solution, which is x = 0 (the zero vector in R

        m) and we call it the trivial solution

        The important question is:

        Does a nontrivial solution exist for a homogeneous linear system ?

        Solution Set of Linear Systems
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        1. For m×n A these are equivalent: Ax = b has a solution for EVERY b ⇔ columns span ℝᵐ ⇔ A has a pivot in every ROW.
        Exam trapSpanning ℝᵐ needs a pivot in every ROW; uniqueness needs a pivot in every COLUMN.
        Slide 7: Do you Remember
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          Slide 8: Slide 8
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          1. Ax = 0 always has the trivial solution x = 0; it has a NONTRIVIAL solution ⇔ there is at least one free variable.
          🎯 How this can come up · 1 question
          MCQ

          A homogeneous system with more unknowns than equations:

          Slide 9: ➔ x & x are basic
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            Slide 10: Example:
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              Slide 11: Slide 11
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                Solution Set of Linear Systems

                Nonhomogeneous Linear Systems: System of linear equations that can be written in the form

                When a nonhomogeneous linear system is consistent, the general solution can be written in parametric vector form as:

                +

                One vector a linear combination of vectors satisfying the corresponding homogeneous system A x = 0.

                Solution Set of Linear Systems
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                1. If Ax = b is consistent with particular solution p, every solution is p + vₕ where vₕ solves Ax = 0 (the solution set is a translated null space).
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