Day 44: Linear Algebra

Today:

• Final Exam: May 9, 10:10 a.m. - 12:10 p.m.
  • The final, 2004
  • The final, 2005

• About grades....

• 6.5 homework returned
  • #21
    1. Ax=0 <=> A'Ax=0, so Nul(A)=Nul(A'A)=0 vector (by 19).
    2. Use the "cartoon form" of A to determine that A must have at least as many rows as columns, because of the independence of the columns.
    3. n

  • Questions about #14, 26
    • #14: norms of Au and Av are bigger than norm(b)
    • If a signal is presented (in the form of a sequence), then the high frequency part will be obliterated (in the null space!), while the lower frequency part will remain (but will be shifted, by the way). This is the idea between filtering in signal processing.

  • Something new, and very important:

• 7.1 homework due: I'll have it back Monday.

• 7.4 homework due with the final.

• Sorry, still no projects graded. I'll have them by Monday, too.

• Current Business:
  • We continue our study of the SVD via the wiki.

  • Another amusing example of SVD compression... (here's the original file)

  • Back to this example:

  • Here is that same matrix, but animated via WebSVD

  • SVD in action - more skinning

  • Here it is in perhaps more detail than we wish to pursue: Section 7.4 highlights (pdf): The Singular Value Decomposition.

• Where have we been? Let's just take a glance at our "index"....

• Evaluations