Diketahui W direntang oleh vektor-vektor berikut: v1 = (1, -1, 2);

Berikut ini adalah pertanyaan dari dzekov08 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Diketahui W direntang oleh vektor-vektor berikut:v1 = (1, -1, 2); v2= (0,1,2); v3= (1,2, -1)

a) Tentukan basis ortonomal W menggunakan proses Gram-Schmidt untuk menentukan basis ortonormal W dengan hasil dalam definisi Euclidean.

b) Cari basis pelengkap orthogonal W

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

Jawab:

a. $W=(v_{1},v_{2},v_{3})$

dengan v1,v2,v3 seperti di soal

$u_{1}=v_{1}=(1,-1,2)\\$

$u_{2}=v_{2}-proj_{u_{1}}(v_{2})\\=v_{2}-\frac{}{||v_{1}||^2}v_{1}\\=(0,1,2)-\frac{0.1+1.-1+2.2}{(\sqrt{1^2+(-1)^2+2^2})^2}(1,-2,2) \\=(0,1,2)-\frac{3}{6}(1,-2,2)\\=(0,1,2)-(\frac{1}{2},-1,1)\\=(-\frac{1}{2},2,1)$

$u_{3}=v_{3}-proj_{u_{1}}(v_{3})-proj_{u_{2}}(v_{3})\\=v_{3}-\frac{}{||v_{1}||^2} v_{1}-\frac{}{||v_{2}||^2} v_{2}\\=(1,2,-1)-\frac{1.1+2(-1)+2(-1)}{6}(1,-1,2)-\frac{1.0+2.1+2(-1)}{5}(0,1,2)\\=(1,2,-1)+(\frac{1}{2},-\frac{1}{2},1 )-(0,\frac{1}{5},\frac{2}{5})\\=(\frac{3}{2},\frac{17}{10},-\frac{2}{5} )$

Basis ortogonal :

$\{u_{1},u_{2},u_{3}\}=\{(1,-1,2),(-\frac{1}{2},2,1 ),(\frac{3}{2},\frac{17}{10},-\frac{2}{5})\}$

Selanjutnya basis ortonormal W :

$q_{1}=\frac{u_{1}}{||u_{1}||} =\frac{(1,-1,2)}{\sqrt{1^2+(-1)^2+2^2} }=\frac{(1,-1,2)}{\sqrt{6} }=(\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}} )$

$q_{2}=\frac{u_{2}}{||u_{2}||} =\frac{(-\frac{1}{2} ,2,1)}{\sqrt{(-\frac{1}{2}) ^2+(2)^2+1^2} }=\frac{(-\frac{1}{2} ,2,1)}{\sqrt{\frac{21}{4} } }=(-\frac{1}{\sqrt{21}},\frac{4}{\sqrt{21} },\frac{2}{\sqrt{21} } )$

$q_{3}=\frac{u_{3}}{||u_{3}||}=\frac{(\frac{3}{2},\frac{17}{10},-\frac{2}{5})}{\sqrt{(\frac{3}{2})^2+(\frac{17}{10})^2+(-\frac{2}{5})^2} }= \frac{(\frac{3}{2},\frac{17}{10},-\frac{2}{5})}{\sqrt{\frac{530}{100}} } =(\frac{15}{\sqrt{530} },\frac{17}{\sqrt{530} },\frac{-4}{\sqrt{530} } )$

b. Pembentukan matriks W :

$W=\left[\begin{array}{ccc}\frac{1}{\sqrt{6} }&-\frac{1}{\sqrt{6}} &\frac{2}{\sqrt{6}}\\-\frac{1}{\sqrt{21}}&\frac{4}{\sqrt{21}}&\frac{2}{\sqrt{21}}\\\frac{15}{\sqrt{530} } &\frac{17}{\sqrt{530} }&-\frac{4}{\sqrt{530} }\end{array}\right]$

ruang null-W adalah pelengkap orthogonal :

$Wx=0\\\left[\begin{array}{ccc}\frac{1}{\sqrt{6} }&-\frac{1}{\sqrt{6}} &\frac{2}{\sqrt{6}}\\-\frac{1}{\sqrt{21}}&\frac{4}{\sqrt{21}}&\frac{2}{\sqrt{21}}\\\frac{15}{\sqrt{530} } &\frac{17}{\sqrt{530} }&-\frac{4}{\sqrt{530} }\end{array}\right]\left[\begin{array}{ccc}a\\b\\c\end{array}\right] =\right]\left[\begin{array}{ccc}0\\0\\0\end{array}\right]\\$

dengan algoritma bareiss didapat :

${\begin{matrix}\frac{-23*\sqrt{1855}}{1113}*a & & & = & 0 \\\frac{-23*\sqrt{1855}}{1113}*b & & = & 0 \\\frac{-23*\sqrt{1855}}{1113}*c & = & 0\end{matrix}$