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Course by: Steven J. Cox. E-mail the author

# 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 NA=NATA 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 ATA A A for the AA matrix associated with the unstable ladder figure.
• (ii) Show that NANATA N A N A A , i.e. that if Ax=0 A x 0 then ATAx=0 A A x 0 .
• (iii) Show that NATANA N A A N A , i.e., that if ATAx=0 A A x 0 then Ax=0 A x 0 . (Hint: if ATAx=0 A A x 0 then xTATAx=0 x A A x 0 .)
3. Suppose that AA is m-by-n and that NA=Rn N A n . Argue that AA must be the zero matrix.

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