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Exercises: Columns and Null Spaces

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

Summary: Exercises for Chapter 3 of CAAM 335, Rice University--Dr. Cox.

Exercises

  1. I encourage you to use rref and null for the following.
    • (i) Add a diagonal crossbar between nodes 3 and 2 in the unstable ladder figure and compute bases for the column and null spaces of the new adjacency matrix. As this crossbar fails to stabilize the ladder, we shall add one more bar.
    • (ii) To the 9 bar ladder of (i) add a diagonal cross bar between nodes 1 and the left end of bar 6. Compute bases for the column and null spaces of the new adjacency matrix.
  2. We wish to show that N⁢A=N⁢AT⁢A N A N A A regardless of AA.
    • (i) We first take a concrete example. Report the findings of null when applied to AA and AT⁢A A A for the AA matrix associated with the unstable ladder figure.
    • (ii) Show that N⁢A⊆N⁢AT⁢A N A N A A , i.e. that if A⁢x=0 A x 0 then AT⁢A⁢x=0 A A x 0 .
    • (iii) Show that N⁢AT⁢A⊆N⁢A N A A N A , i.e., that if AT⁢A⁢x=0 A A x 0 then A⁢x=0 A x 0 . (Hint: if AT⁢A⁢x=0 A A x 0 then xT⁢AT⁢A⁢x=0 x A A x 0 .)
  3. Suppose that AA is m-by-n and that N⁢A=Rn N A n . Argue that AA must be the zero matrix.

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