Question

1 4 3 13 The vectors V1 = | 2 and V2 = 5 span a subspace V of the indicated Euclidean space. Find a basis for the orthogonal complement vt of V. 8 36 4 13 Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. O A. A basis for the orthogonal complement vt is {}. (Use a comma to separate vectors as needed.) OB. There is no basis for the orthogonal complement v+.

Answer 1

write both vectors in the matrix

$A=\begin{pmatrix}1&3&2&8&4\\ 4&13&5&36&13\end{pmatrix}$

$R_1\:\leftrightarrow \:R_2$

$=\begin{pmatrix}4&13&5&36&13\\ 1&3&2&8&4\end{pmatrix}$

$R_2\:\leftarrow \:R_2-\frac{1}{4}\cdot \:R_1$

$=\begin{pmatrix}4&13&5&36&13\\ 0&-\frac{1}{4}&\frac{3}{4}&-1&\frac{3}{4}\end{pmatrix}$

$R_2\:\leftarrow \:-4\cdot \:R_2$

$=\begin{pmatrix}4&13&5&36&13\\ 0&1&-3&4&-3\end{pmatrix}$

$R_1\:\leftarrow \:R_1-13\cdot \:R_2$

$=\begin{pmatrix}4&0&44&-16&52\\ 0&1&-3&4&-3\end{pmatrix}$

$R_1\:\leftarrow \frac{1}{4}\cdot \:R_1$

$=\begin{pmatrix}1&0&11&-4&13\\ 0&1&-3&4&-3\end{pmatrix}$

reduced system is

$\begin{pmatrix}1&0&11&-4&13\\ 0&1&-3&4&-3\end{pmatrix}\begin{pmatrix}a\\ b\\ c\\ d\\ e\end{pmatrix}=0$

$a+11c-4d+13e=0....................a=-11c+4d-13e$

$b-3c+4d-3e=0..................b=3c-4d+3e$

$c=c...............free$

$d=d...............free$

$e=e...............free$

general solution is

$\begin{pmatrix}a\\ b\\ c\\ d\\ e\end{pmatrix}=\begin{pmatrix}-11\\ 3\\ 1\\ 0\\ 0\end{pmatrix}c+\begin{pmatrix}4\\ -4\\ 0\\ 1\\ 0\end{pmatrix}d+\begin{pmatrix}-13\\ 3\\ 0\\ 0\\ 1\end{pmatrix}e$

basis for orthogonal complement is

${\color{Red} V^{\perp}:\left \{ \begin{pmatrix}-11\\ 3\\ 1\\ 0\\ 0\end{pmatrix},\:\begin{pmatrix}4\\ -4\\ 0\\ 1\\ 0\end{pmatrix},\:\begin{pmatrix}-13\\ 3\\ 0\\ 0\\ 1\end{pmatrix} \right \}}$