Linear Algebra with Applications, 10th Edition

5.3

1. 1. 

   1. (a) ${(2,1)}^T$;

   2. (c) ${(1.6,0.6,1.2)}^T$

2. 2. 

   1. (1a) $\mathbf{p}={(3,1,0)}^T,\mathbf{r}=(0,0,2{)}^T$

   2. (1c) $\mathbf{p}={(3.4,0.2,0.6,2.8)}^T,$

      $\mathbf{r}={(0.6,-0.2,0.4,-0.8)}^T$

3. 3. 

   1. (a) $\{{(1-2\alpha ,\alpha )}^T|\alpha \text{ real}\}$;

   2. (b) $\{{(2-2\alpha ,1-\alpha ,\alpha )}^T|\alpha \text{ real}\}$

4. 4. 

   1. (a) $\mathbf{p}={(1,2,-1)}^T,\mathbf{b}-\mathbf{p}={(2,0,2)}^T$;

   2. (b) $\mathbf{p}={(3,1,4)}^T,\mathbf{p}-\mathbf{b}={(-5,-1,4)}^T$

5. 5. 

   1. (a) $y=1.8+2.9x$

6. 6. $0.55+1.65x+1.25x^2$

7. 9.  

   1. (a) Hint: If $\mathbf{\text{b}}\in R\left(a\right)$ then $\mathbf{\text{b =}}A\mathbf{\text{x}}$ for some $\mathbf{\text{x}}\in {ℝ}^n$.

   2. (b) Hint: ${R\left(A\right)}^{\perp }=N\left(A^T\right)$.

8. 10. Hint: If the case of a consistent system, what can we conclude about b in relation to one of the fundamental subspaces?

9. 14. The least squares circle will have center $(0.58,-0.64)$ and radius 2.73 (answers rounded to two decimal places).

10. 15. 

    1. (a) $\mathbf{w}={(0.1995,0.2599,0.3412,0.1995)}^T$

    2. (b) $\mathbf{r}={(0.2605,0.2337,0.2850,0.2208)}^T$