Linear Algebra with Applications, 10th Edition

3.6

1. 2. 

   1. (a) 3;

   2. (b) 3;

   3. (c) 2

2. 3. 

   1. (a) ${\mathbf{u}}_2,{\mathbf{u}}_4,{\mathbf{u}}_5$ are the column vectors of U corresponding to the free variables. ${\mathbf{u}}_2=2{\mathbf{u}}_1,{\mathbf{u}}_4=5{\mathbf{u}}_1-{\mathbf{u}}_3,{\mathbf{u}}_5=-3{\mathbf{u}}_1+2{\mathbf{u}}_3$

   2. (b) Hint: The column vectors of A satisfy the same dependency relations that the column vectors of U satisfy.

3. 4. 

   1. (a) consistent;

   2. (b) inconsistent;

   3. (e) consistent

4. 5. 

   1. (a) infinitely many solutions;

   2. (c) unique solution

5. 6. Hint: Use the consistency theorem to determine if the system is consistent. If A is an $m\times n$ matrix with linearly dependent columns, then what can you conclude about its nullity? For a consistent system what does the nullity of the coefficient matrix tell you about the number of solutions?

6. 7. 

   1. (b) Hint: If A has rank 6, then the column vectors of A will span ${ℝ}^6\mathrm{.}$

7. 8.  

   1. (a) Hint: If ...