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Back to Gaussian elimination revisited

Exercise 10

Avg difficulty 2.5

Consider the matrix A=[−11001−21001−2100−11].A = \left[\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & -1 & 1 \\ \end{array}\right]\text{.}

  1. Find the LULU factorization of AA.

  2. Use the factorization to find a basis for Nul(A)\nul(A).

  3. We have seen that Nul(A)=Nul(U)\nul(A) = \nul(U). Is it true that Col(A)=Col(L)\col(A) = \col(L)?

Austin, Understanding Linear Algebra, §5.1 Gaussian elimination revisited, Exercise 10

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.