1. Echelon forms.
Describe row echelon form and reduced row echelon form in your own words.
Reflections 2.7 Reflect on your understanding
In addition to the reflection activities below, re-read Section 2.2 Terminology and notation. Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
2. Reducing matrices.
Write a simplified version of Procedure 2.3.2 in your own words.
3. Determining the number of solutions.
Describe the different patterns in reduced augmented matrices that determine whether the associated system has no solution, one unique solution, or an infinite number of solutions. Further describe why each pattern tells the number of solutions (none/one/infinitely many).
4. Number of solutions for a homogeneous systems.
(a)
Explain why it is true that every homogeneous system is consistent.
(b)
Explain why a homogeneous system with fewer equations than variables must have at least one nontrivial solution.
5. Rank versus parameters.
(a)
Describe the relationship between the rank of a matrix and the number of parameters required to express the general solution to the homogeneous system for which that matrix is the coefficient matrix.
(b)
In particular, what must the rank of a matrix be for the corresponding homogeneous system to have only the trivial solution?
6. Row equivalence to RREF.
(a)
Suppose two students are given the same matrix to reduce, and they choose different sequences of row operations. Assume they perform all operations correctly, and both eventually arrive at a matrix in RREF.
(i)
Even though they performed different sequences of operations, could their RREF results be the same as each other?
(ii)
Since they performed different sequences of operations, could their RREF results be different from each other?
(b)
Suppose two students are given different matrices to reduce. Again assume they perform all operations correctly, and both eventually arrive at a matrix in RREF.
Even though they started with different matrices, could their RREF results be the same as each other?
(c)
Is it possible for two different matrices in REF to be row equivalent?
