91Ó°ÊÓ

Prove that if \(S_{1}\) and \(S_{2}\) are orthogonal subspaces of \(R^{n},\) then their intersection consists only of the zero vector.

Find an orthonormal basis for the solution space of the homogeneous system of linear equations. $$\begin{aligned} 2 x_{1}+x_{2}-6 x_{3}+2 x_{4} &=0 \\ x_{1}+2 x_{2}-3 x_{3}+4 x_{4} &=0 \\ x_{1}+x_{2}-3 x_{3}+2 x_{4} &=0 \end{aligned}$$

(a) find the linear least squares approximating function \(g\) for the function \(f\) and \((b)\) use a graphing utility to graph \(f\) and \(g\). $$f(x)=\sqrt{x}, 1 \leq x \leq 4$$

Find an orthonormal basis for the solution space of the homogeneous system of linear equations. $$\begin{array}{l} x_{1}-x_{2}+x_{3}+x_{4}=0 \\ x_{1}-2 x_{2}+x_{3}+x_{4}=0 \end{array}$$