2. 

1. (a) 3;

2. (b) 3;

3. (c) 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.

4. 

1. (a) consistent;

2. (b) inconsistent;

3. (e) consistent

5. 

1. (a) infinitely many solutions;

2. (c) unique solution

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?

7. 

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

8.  

1. (a) Hint: If $N\left(A\right)=\left\{\mathbf{0}\right\},$ then what will the rank of A be?

2. (b) Hint: Hint. Use the consistency theorem. If a system is consistent and $N\left(A\right)=\left\{\mathbf{0}\right\},$ then what can you conclude about the number of solutions? Alternatively, Theorem 3.3.2 could be helpful in answering these questions.

10. 

1. (b) Hint: Whenever you represent a vector as a linear combination of linearly independent vectors, that representationwill be unique.

12. 

1. (a) Hint: If A and B are row equivalent, then how are their ranks related?

2. (b) Hint: Look at some examples. Make up a singular matrix A whose entries are all nonzero and then reduce A to its echelon form U. Do U and A have the same column spaces?

14. Hint: The column vectors of A satisfy the same dependency relations that the column vectors of U satisfy.

15. 

1. (b) (ii). Hint: Use the equation $A{\mathbf{\text{x}}}_0=\mathbf{\text{b}}$ to solve for ${\mathbf{\text{a}}}_4\mathrm{.}$

16. Hint: Show that the system must be consistent and that the echelon form of the coefficient matrix will involve free variables.

18. 

1. (b) $n-1$

19. Hint: $\mathbf{\text{y}}=A\mathbf{\text{x}}\ne \mathbf{0}$ is equivalent to saying that x is not in the null space of A.

21. Hint: If you have bases for the column spaces of A and B, you can use them to form a spanning for the column space of $A+B\mathrm{.}$

22. 

1. (a) Hint: Show that if $\mathbf{\text{x}}\in N\left(A\right)$ then $\mathbf{\text{x}}\in N\left(BA\right)$ and show the converse, if $\mathbf{\text{x}}\in N\left(BA\right)$ then $\mathbf{\text{x}}\in N\left(A\right)\mathrm{.}$

2. (b) Hint: Use part (a) to show first that $(AC{)}^T$ and $A^T$ have the same rank.

24. Hint: What is the rank of $A-B?$

25. 

1. (b) Hint: Partition B into columns and perform the block multiplication

   $$AB=A({\mathbf{b}}_1,{\mathbf{b}}_2,\mathrm{...},{\mathbf{b}}_n)=(A{\mathbf{b}}_1,A{\mathbf{b}}_2,\mathrm{...},A{\mathbf{b}}_n)$$

27. 

1. (b) Hint: Use the Rank-Nullity theorem.

28. 

1. (a) Hint: Show that each column vector of C is a linear combination of the column vectors of A.

2. (b) Hint: $C^T=B^T{A}^T\mathrm{.}$

29. 

1. (a) Hint: In general a matrix E will have linearly independent column vectors if and only if $E\mathbf{\text{x}}=\mathbf{0}$ has only the trivial solution $\mathbf{\text{x}}=\mathbf{0.}$ One way to show that C has linearly independent column vectors is to show that $C\mathbf{\text{x}}\ne \mathbf{0}$ for all $\mathbf{\text{x}}\ne \mathbf{0}$ and hence that $C\mathbf{\text{x}}=\mathbf{0}$ has only the trivial solution.

2. (b) Hint: $C^T=B^T{A}^T\mathrm{.}$

30. 

1. (a) Hint: If the column vectors of B are linearly dependent then $B\mathbf{\text{x}}=\mathbf{0}$ for some nonzero vector $\mathbf{\text{x}}\in {ℝ}^r\mathrm{.}$

31. 

1. (a) Hint: To get started, if C is a right inverse of A, let b be any vector in ${ℝ}^m$ and let $\mathbf{c}=C\mathbf{b}\mathrm{.}$

2. (b) Hint: If n vectors span ${ℝ}^m$ then how must m and n be related?

32. If ${\mathbf{x}}_j$ is a solution to $A\mathbf{x}={\mathbf{e}}_j$ for $j=1,\dots ,m$ and $X=({\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_m)$ then $\mathit{\text{AX}}=I_m$.

34. Hint: Let B be an $n\times m$ matrix. If B has a left inverse, then $B^T$ has a right inverse. Apply the result of Exercise 31 to $B^T\mathrm{.}$