Linear Algebra, 5th Edition

Section 5.1

1. 1.

   1. (a) F

   2. (b) T

   3. (c) T

   4. (d) F

   5. (e) F

   6. (f) F

   7. (g) F

   8. (h) T

   9. (i) T

   10. (j) F

   11. (k) F

2. 2.

   1. (a) $11,t^2-5t+11$

   2. (c) $-2,t^4-2t^3+t^2-2$

3. 3.

   1. (a) ${[\text{T}]}_{\beta }=\left(\begin{array}{rr}0& 2\\ -1& 0\end{array}\right),\text{ no}$

   2. (c) ${[\text{T}]}_{\beta }=\left(\begin{array}{rrr}-1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right),\text{ yes}$

   3. (e) ${[\text{T}]}_{\beta }=\left(\begin{array}{rrrr}-1& 1& 0& 0\\ 0& -1& 1& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),\text{ no}$

4. 4.

   1. (a) The eigenvalues are 4 and $-1,$ a basis of eigenvectors is $\left\{\left(\begin{array}{l}2\\ 3\end{array}\right),\left(\begin{array}{r}1\\ -1\end{array}\right)\right\},Q=\left(\begin{array}{rr}2& 1\\ 3& -1\end{array}\right),$ and $D=\left(\begin{array}{rr}4& 0\\ 0& -1\end{array}\right)\text{.}$

   2. (c) The eigenvalues are 1 and $-1,$ a basis of eigenvectors is $\left\{\left(\begin{array}{c}1\\ 1-i\end{array}\right),\left(\begin{array}{c}1\\ -1-i\end{array}\right)\right\},Q=\left(\begin{array}{cc}1& 1\\ 1-i& -1-i\end{array}\right),$ and $D=\left(\begin{array}{rr}1& 0\\ 0& -1\end{array}\right)\text{.}$

5. 5.

   1. (a) $\lambda =3,4\beta =\{(3,5),(1,2)\}$

   2. (b) $\lambda =-1,1,2\beta =\{(1,2,0),(1,-1,-1),(2,0,-1)\}$

   3. (f) $\lambda =1,3\beta =\{-2+x,-4+x^2,-8+x^3,x\}$

   4. (h) $\lambda =-1,1,1,1\beta =\left\{\left(\begin{array}{ll}-1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{ll}0& 1\\ 0& 0\end{array}\right),\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{ll}0& 0\\ 1& 0\end{array}\right)\right\}$

   5. (i) $\lambda =1,1,-1,-1\beta =\left\{\left(\begin{array}{rr}1& 0\\ 1& 0\end{array}\right),\left(\begin{array}{rr}0& 1\\ 0& 1\end{array}\right),\left(\begin{array}{rr}-1& 0\\ 1& 0\end{array}\right),\left(\begin{array}{rr}0& -1\\ 0& 1\end{array}\right)\right\}$

   6. (j) $\lambda =-1,1,5\beta =\left\{\left(\begin{array}{rr}0& 1\\ -1& 0\end{array}\right),\left(\begin{array}{rr}1& 0\\ 0& -1\end{array}\right),\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right),\left(\begin{array}{rr}1& 0\\ 0& 1\end{array}\right)\right\}$

6. 25. 4