MATH 2121: Lecture 4

Date: 2024-09-12 15:03:35

Reviewed:

Topic / Chapter: Matrix Equations

summary

❓Questions

Notes

Multiplying Matrices and Vectors

  • Matrix as vector operator

    • given

      • and

      • then matrix-vector product

      • and : linear combination on col. of where coeff. are provided by entries

    • example
    • if , then only defined for
      • i.e. no. of columns of should match size (rows) of
      • then
    • : transforms to

  • Linear-ness of transformation

    • transformation is linear
      1. if and then
      2. if and and then
    • line: a spane of one vector
      • then
    • e.g. : counterclockwise rotation, 90 degrees
Matrix Equations

  • Recap

    • linear system: can be written as vector equation as well
    • why not matrix?

  • Matrix equation

    • when ,
    • where are each variable
      • : matrix equation
    • : has same solution as
      • and lin. sys. with augmented matrix
    • : has solution being lin. combination of columns of

  • Linear systems trilogy

    • for matrix , following properties are equivalent
      1. for each vector , the matrix has a solution
      2. each vector is a linear combination fo the columns of
      3. the span of the columns of is the set (read: "columns of span ")
        • same to (2), but slightly different wordings
      4. has a pivot position in every row
        • not intuitive, but computable
    • proof:
      • 1-3: different way of saying the same thing
      • 4: if has pivot in every row
        • cannot have a pivot position in the last column
        • i.e.
Linear independence

  • Linear independence

    • are linearly independent if the only solution to
      • is
      • is there is any non-zero solution: vectors are linearly independent
    • theorem: The columns of a matrix A are linearly independent if and only if A has a pivot position in every column
    • theorem: Suppose . If then these vectors are linearly dependent
      • : no. of cols.