1. Let a1, a2, a3 be linearly independent vectors in R3, and let A = [a1 a2 a3]. Which of the following statements are true?
a) The reduced row echelon form of A is I3
b) The rank of A is 3
c) The system [A|b] has a unique solution for any vector b in R3
d) (a), (b) and (c) are all true
e) (a) and (b) are both true, but not (c)
2. Show that span (u,v) = span (u, u+v) for any vectors u and v.
3. If A is a diagonalisable matrix and all of its eigenvalues satisfy |λ| < 1, prove that An approaches the zero matrix as n gets large.
4. Prove that given any n real numbers λ1, ..., λn, there exists a symmetric n x n matrix with λ1, ..., λn as its eigenvalues.
5. Find the LU factorisation of the matrix shown below:
6. Write the given permutation matrix shown below; as a product of elementary (row interchange) matrices.
This is a series of linear algebra problems that involve span and LU factoring