Subject: Applied Mathematics 4

Topic: Vector Spaces

Difficulty: Medium

Let u$_1$ = (3,1) and u$_2$ = (4,2)

Let $v_1$ = u$_1$ = (3,1)

$ v_2 = u_2 - [\frac{\lt u_2,v_1 \gt}{||v||^2}]v_1 \\ = (4,2) - [\frac{(4,2).(3,1)}{\sqrt{9+1}^2}](3,1) \\ = (4,2) - [\frac{12+2}{\sqrt{10}^2}](3,1) \\ = (4,2) - [\frac{14}{10}](3,1) \\ = (4,2) - [\frac{7}{5}](3,1) \\ = (4,2) - [\frac{21}{5},\frac{7}{5}] \\ = (4 - \frac{21}{5}, 2 - \frac{7}{5} \\ = (\frac{-1}{5},\frac{3}{5}) \\ ||v_1|| = \sqrt{10} \\ ||v_2|| = \sqrt{\frac{1}{25} + \frac{9}{25}} = \sqrt{\frac{10}{25}} = \frac{\sqrt{2}}{\sqrt{5}} \\ q_1 = \frac{v_1}{||v_1||} = (\frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}) \\ q_2 = \frac{v_2}{||v_2||} = \frac{-1/5,3/5}{\sqrt{2}/\sqrt{5}} = \frac{\sqrt{5}}{\sqrt{2}}(\frac{-1}{5},\frac{3}{5}) = \frac{-1}{\sqrt{10}},\frac{3}{\sqrt{10}})$

Hencem the orthogonal basis is {$ q_1,q_2 $}