3.6: Exercises- Columns and Null Spaces

Exercise \(\PageIndex{1}\)

I encourage you to use rref and null for the following.

1. 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.
2. 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.

Exercise \(\PageIndex{2}\)

We wish to show that \(N(A)=N(A^{T}A)\) regardless of \(A\).

1. We first take a concrete example. Report the findings of null when applied to \(A\) and \(A^{T}A\) for the \(A\) matrix associated with the unstable ladder figure.
2. Show that \(N(A) \subseteq N(A^{T}A)\) i.e. that if \(A \textbf{x} = \textbf{0}\) then \(A^{T}A \textbf{x} = \textbf{0}\).
3. Show that \(N(A^{T}A) \subseteq N(A)\) i.e., that if \(A^{T}A \textbf{x} = \textbf{0}\) then \(A \textbf{x} = \textbf{0}\) (Hint: if \(A^{T}A \textbf{x} = \textbf{0}\) then \(\textbf{x}^{T} A^{T}A \textbf{x} = \textbf{0}\)

Exercise \(\PageIndex{3}\)

Suppose that \(A\) is m-by-n and that \(N(A) = \mathbb{R}^{n}\). Argue that \(A\) must be the zero matrix.