Exam 6: Orthogonality and Least Squares

Given A and b, determine the least-squares error in the least-squares solution of Ax = b. - $A = \left[ \begin{array} { l l } 4 & 3 \\2 & 1 \\3 & 2\end{array} \right] , b = \left[ \begin{array} { l } 3 \\0 \\1\end{array} \right]$

Write the word or phrase that best completes each statement or answers the question. -As a function, $\mathrm { q } _ { 3 } ( \mathrm { t } ) = 12 \mathrm { t } ^ { 2 } - 12 \mathrm { t } + 2$ . The orthogonal basis for the subspace $\mathrm { W }$ is $\left\{ q _ { 1 } , q _ { 2 } , q _ { 3 } \right\}$ .

Find a unit vector in the direction of the given vector. - $\left[ \begin{array} { r } - 4 \\ 4 \\ - 2 \end{array} \right]$

Write the word or phrase that best completes each statement or answers the question. -For $f , g$ in C[a, b], set $\langle f , g \rangle = \int _ { a } ^ { b } f ( t ) g ( t ) d t$ . Show that $\langle f , g \rangle$ defines an inner product of $C [ a , b ]$ .

$\text { Find the least-squares line } y = \beta _ { 0 } + \beta _ { z } x \text { that best fits the given data. }$ -Given: The data points (-3, 2), (-2, 5), (0, 5), (2, 3), (3, 3). Suppose the errors in measuring the y-values of the last two data points are greater than for the Other points. Weight these data points half as much as the rest of the data. $X = \left[ \begin{array} { r r } 1 & - 3 \\1 & - 2 \\1 & 0 \\1 & 2 \\1 & 3\end{array} \right] , \beta = \left[ \begin{array} { l } \beta _ { 1 } \\\beta _ { 2 }\end{array} \right] , y = \left[ \begin{array} { l } 2 \\5 \\5 \\3 \\3\end{array} \right]$

Let W be the subspace spanned by the uʹs. Write y as the sum of a vector in W and a vector orthogonal to W. - $\mathbf { y } = \left[ \begin{array} { r } 19 \\3 \\11\end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 1 \\0 \\- 1\end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { l } 2 \\1 \\2\end{array} \right]$

Find the distance between the two vectors. - $\mathbf { u } = ( 0,0,0 ) , \mathbf { v } = ( - 5 , - 3 , - 4 )$

Determine whether the set of vectors is orthogonal. - $\left[ \begin{array} { l } - 20 \\- 40 \\- 20\end{array} \right] , \left[ \begin{array} { r } 20 \\0 \\- 20\end{array} \right] , \left[ \begin{array} { l } - 20 \\- 20 \\- 20\end{array} \right]$

Compute the length of the given vector. - $\mathrm { p } ( \mathrm { t } ) = 13 \mathrm { t } ^ { 2 }$ and $\mathrm { q } ( \mathrm { t } ) = \mathrm { t } - 1$ , where $\mathrm { t } _ { 0 } = 0 , \mathrm { t } _ { 1 } = \frac { 1 } { 2 } , \mathrm { t } _ { 2 } = 1$

Compute the dot product u · v. - $\mathbf { u } = \left[ \begin{array} { r } 10 \\0\end{array} \right] , \mathbf { v } = \left[ \begin{array} { c } 14 \\- 3\end{array} \right]$

$\text { Find the least-squares line } y = \beta _ { 0 } + \beta _ { z } x \text { that best fits the given data. }$ -Given: The data points (-2, 2), (-1, 5), (0, 5), (1, 3), (2, 5). 44) Suppose the errors in measuring the y-values of the last two data points are greater than for the Other points. Weight these data points twice as much as the rest of the data. $\mathrm { X } = \left[ \begin{array} { r r } 1 & - 2 \\1 & - 1 \\1 & 0 \\1 & 1 \\1 & 2\end{array} \right] , \beta = \left[ \begin{array} { l } \beta _ { 1 } \\\beta _ { 2 }\end{array} \right] , \mathrm { y } = \left[ \begin{array} { l } 2 \\5 \\5 \\3 \\5\end{array} \right]$

Find the distance between the two vectors. - $\mathbf { u } = ( 24,14,19 ) , \mathbf { v } = ( - 4,4,7 )$

Let W be the subspace spanned by the uʹs. Write y as the sum of a vector in W and a vector orthogonal to W. - $\mathbf { y } = \left[ \begin{array} { r } 17 \\ 7 \\ 12 \end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right]$

Express the vector x as a linear combination of the uʹs. - $\mathbf { u } _ { 1 } = \left[ \begin{array} { r } 2 \\- 4\end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r } 12 \\6\end{array} \right] , \mathbf { x } = \left[ \begin{array} { c } 32 \\- 4\end{array} \right]$

Solve the problem. -Let $C [ 0 , \pi ]$ have the inner product $\langle f , g \rangle = \int _ { 0 } ^ { \pi } f ( t ) g ( t ) d t$ , and let $m$ and $n$ be unequal positive integers. Prove that $\cos ( \mathrm { mt } )$ and $\cos ( \mathrm { nt } )$ are orthogonal.

Compute the dot product u · v. - $\mathbf { u } = \left[ \begin{array} { r } 1 \\17\end{array} \right] , \mathbf { v } = \left[ \begin{array} { l } 3 \\1\end{array} \right]$

Find the closest point to y in the subspace W spanned by u₁ and u₂. - $\mathbf { y } = \left[ \begin{array} { r } 16 \\0 \\2\end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 1 \\0 \\- 1\end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { l } 2 \\1 \\2\end{array} \right]$

Find the distance between the two vectors. - $\mathbf { u } = ( 6 , - 12 ) , \mathbf { v } = ( - 12,12 )$

Find a unit vector in the direction of the given vector. - $\left[ \begin{array} { r } 13 \\ - 26 \end{array} \right]$

Find the orthogonal projection of y onto u. - $\mathbf { y } = \left[ \begin{array} { r } - 12 \\36\end{array} \right] , \mathbf { u } = \left[ \begin{array} { r } - 8 \\4\end{array} \right]$