Numerical Methods & Linear Algebra

Math 2890-003

(1 point) Let \[u=\left(\begin{array}{r} -1 \\ 1 \\ -7 \\ 8 \end{array}\right)\quad\text{and}\quad v = \left(\begin{array}{r} 4 \\ -6 \\ 7 \\ -7 \end{array}\right).\] Find the inner product $u \cdot v$. Show your work.

(1 point) Let \[u=\left(\begin{array}{r} -1 \\ -3 \\ -5 \\ 0 \end{array}\right)\quad\text{and}\quad v = \left(\begin{array}{r} 7 \\ -3 \\ -4 \\ -1 \end{array}\right)\]. Find the inner product $u \cdot v$. Show your work.

(1 point) Let \[u=\left(\begin{array}{r} 4 \\ -7 \\ 6 \end{array}\right)\quad\text{and}\quad v = \left(\begin{array}{r} 3 \\ 5 \\ 4 \end{array}\right).\] Find the distance between $u$ and $v$. Show and explain your computations.

(1 point) Let \[u_1=\left(\begin{array}{r} 5 \\ -1 \\ 1 \\ -5 \end{array}\right),\quad u_2=\left(\begin{array}{r} 2 \\ 8 \\ 8 \\ 2 \end{array}\right)\quad\text{and}\quad u_3=\left(\begin{array}{r} 7 \\ -7 \\ 3 \\ 9 \end{array}\right).\] Is the set $\{u_1, u_2, u_3 \}$ orthogonal? Why or why not? Show your computations.

(1 point) Let \[y=\left(\begin{array}{r} -5 \\ 2 \\ -5 \end{array}\right)\] and let $W$ be the span of \[\left(\begin{array}{r} 1 \\ 0 \\ 2 \end{array}\right) \text{ and } \left(\begin{array}{r} 0 \\ -2 \\ -2 \end{array}\right).\] Project $y$ onto $W$. Show and explain your computations.

(1 point) Let \[y=\left(\begin{array}{r} 1 \\ -1 \\ -7 \end{array}\right)\] and let $W$ be the span of \[\left(\begin{array}{r} 1 \\ -5 \\ -4 \end{array}\right) \text{ and } \left(\begin{array}{r} 2 \\ -8 \\ 2 \end{array}\right).\] Find the point in $W$ that is closest to $y$. Show and explain your computations.

(1 point) Let \[y=\left(\begin{array}{r} 5 \\ 4 \\ 2 \\ -2 \end{array}\right)\] and let $W$ be the span of \[\left(\begin{array}{r} -2 \\ 2 \\ 0 \\ -2 \end{array}\right) \text{ and } \left(\begin{array}{r} 3 \\ -8 \\ -5 \\ -7 \end{array}\right).\] Write $y$ as a sum of a vector in $W$ and a vector orthogonal to $W$. Show and explain your computations.

(1 point) Let \[A=\left(\begin{array}{rr} -1 & -2 \\ 0 & 2 \\ 2 & 6 \\ 2 & 2 \end{array}\right)\quad\text{and}\quad b=\left(\begin{array}{r} -4 \\ 1 \\ -5 \\ 1 \end{array}\right).\] Find the least squares solution to $Ax=b$. Show and explain your computations.

(1 point) Let \[A=\left(\begin{array}{rr} 1 & -5 \\ 4 & -1 \\ -2 & -1 \\ 1 & -2 \end{array}\right)\quad\text{and}\quad b=\left(\begin{array}{r} 3 \\ 3 \\ 2 \\ -2 \end{array}\right).\] Find the least squares error in the least squares solution to $Ax=b$. Show and explain your computations.
Hint: The least squares solution is $x=\left(\begin{array}{r} 0.2246 \\ -0.4509 \end{array}\right)$.

(1 point) Use the QR factorizaton \begin{align*}A&=\left(\begin{array}{rr} -0.199 & 0.4302 \\ 0 & -1.5113 \\ 1.5921 & 2.6036 \\ 1.194 & 4.9752 \end{array}\right)\\ \\ &= \underbrace{\left(\begin{array}{rr} -0.0995 & 0.3092 \\ 0 & -0.5038 \\ 0.796 & -0.4589 \\ 0.597 & 0.6634 \end{array}\right)}_Q \underbrace{\left(\begin{array}{rr} 2 & 5 \\ 0 & 3 \end{array}\right)}_R.\end{align*} to find the least squares solution to $Ax=b$, where $b=\left(\begin{array}{r} -4 \\ 0 \\ -4 \\ 5 \end{array}\right)$. Show your work.

(1 point) Let \[A=\left(\begin{array}{rrr} -2 & -4 & 11 \\ -4 & -13 & 17 \\ 5 & 15 & -18 \end{array}\right).\] Find the QR factorization of $A$. Show and explain your computations.

(1 point) Use the QDR factorization\begin{align*}A&=\left(\begin{array}{rrr} -7 & 29 & -36 \\ 3 & -11 & 6 \\ 3 & -11 & 14 \\ 2 & -9 & 7 \\ 3 & -11 & -2 \end{array}\right)\\ &=\underbrace{\left(\begin{array}{rrr} -7 & 1 & -3 \\ 3 & 1 & -1 \\ 3 & 1 & 7 \\ 2 & -1 & -6 \\ 3 & 1 & -9 \end{array}\right)}_Q \underbrace{\left(\begin{array}{rrr} 1/80 & 0 & 0 \\ 0 & 1/5 & 0 \\ 0 & 0 & 1/176 \end{array}\right)}_D \underbrace{\left(\begin{array}{rrr} 80 & -320 & 320 \\ 0 & 5 & -25 \\ 0 & 0 & 176 \end{array}\right)}_R \end{align*}to find the least squares solution to $Ax=b$ where $b=\left(\begin{array}{r} -5 \\ 5 \\ -4 \\ 5 \\ -4 \end{array}\right)$. Show your work.

(1 point) Let \[A=\left(\begin{array}{rrr} -1 & 2 & -8 \\ 3 & -6 & 20 \\ -5 & 15 & -11 \\ -2 & 9 & 13 \\ 1 & -7 & -17 \end{array}\right).\] Find the QDR factorization of $A$. Show and explain your computations.

(1 point) Consider the data points $(1,2),(2,-2),(3,-6),(4,3),(5,5)$.
Find the equation $y=\beta_0 + \beta_1 x$ of the least-squares line that best fits the given data points. Show and explain your computations.

(1 point) Consider the data points $(1,1),(2,-6),(3,4),(4,-9),(5,2)$.
Find the equation $y=\beta_0 + \beta_1 x + \beta_2 x^2$ of the least-squares quadratic that best fits the given data points. Show and explain your computations.