91Ó°ÊÓ

Show that the function defines an inner product on \(R^{2},\) where \(\mathbf{u}=\left(u_{1}, u_{2}\right)\) and \(\mathbf{v}=\left(v_{1}, v_{2}\right)\) $$\langle\mathbf{u}, \mathbf{v}\rangle= u_{1} v_{1}+9 u_{2} v_{2}$$

Finding the Orthogonal Complement and Direct Sum In Exercises \(9-14,\) (a) find the orthogonal complement \(S^{2},\) and \((b)\) find the direct \(\operatorname{sum} S \oplus S^{+}\) $$ S=\operatorname{span}\left\\{\left[\begin{array}{r} 0 \\ -2 \\ 1 \end{array}\right]\right\\} $$

(a) determine whether the set of vectors in \(R^{n}\) is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for \(R^{n}\). \(\\{(-6,3,2,1),(2,0,6,0)\\}\)

Find bases for the four fundamental subspaces of the matrix \(A\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 0 \end{array}\right] $$

Find bases for the four fundamental subspaces of the matrix \(A\). $$ A=\left[\begin{array}{rrr} 0 & -1 & 1 \\ 1 & 2 & 0 \\ 1 & 1 & 1 \end{array}\right] $$

Find the least squares regression quadratic polynomial for the data points. $$ (0,2),\left(1, \frac{3}{2}\right),\left(2, \frac{5}{2}\right),(3,4) $$

Apply the Gram-Schmidt orthonormalization process to transform the given basis for a subspace of \(\boldsymbol{R}^{n}\) into an orthonormal basis for the subspace. Use the vectors in the order in which they are given. \(B=\\{(3,4,0),(2,0,0)\\}\)

Determine whether \(u\) and \(v\) are orthogonal, parallel, or neither. $$\mathbf{u}=\left(-\frac{1}{3}, \frac{2}{3}\right), \quad \mathbf{v}=(2,-4)$$

Apply the alternative form of the Gram-Schmidt orthonormalization process to find an orthonormal basis for the solution space of the homogeneous linear system. \(\begin{aligned} x_{1}+x_{2}-x_{3}-x_{4} &=0 \\ 2 x_{1}+x_{2}-2 x_{3}-2 x_{4} &=0 \end{aligned}\)

Determine whether \(u\) and \(v\) are orthogonal, parallel, or neither. $$\mathbf{u}=\left(4, \frac{3}{2},-1, \frac{1}{2}\right), \quad \mathbf{v}=\left(-2,-\frac{3}{4}, \frac{1}{2},-\frac{1}{4}\right)$$