Prof. A. Suciu

MTH 1230 -- Linear Algebra -- Fall 1997

Quiz 4

(8 points) Let and .
- Find the least squares solution $\bigl[ \begin{smallmatrix} \overline{x}\\ \overline{y}\end{smallmatrix}\bigr]$ of the inconsistent system $A\cdot \bigl[ \begin{smallmatrix} x\\ y\end{smallmatrix}\bigr] = \mathbf{b}$ .
- Use your answer to part one to find the projection of $\mathbf{b}$ onto $\CS(A)$ , the column space of A.
(8 points) A company gathers the following data:
$\begin{displaymath} \begin{tabular} {\vert p{9pc}\vert\vert c\vert c\vert c\vert... ...l (in millions of dollars)}&0.8&2&3&4.2&5\\ \hline\end{tabular}\end{displaymath}$

Represent the years 1993, ... ,1997 as -2,-1,0,1,2, respectively, and let x denote the year. Let y denote the annual sales (in millions of dollars).
- Find the least squares line relating x and y.
- Use the equation obtained in part a to estimate the annual sales for the year 1999.
(8 points) Apply the Gram-Schmidt process to the vectors $\mathbf{a}_1=\bmatrix 1\\ -1 \endbmatrix$ , $\mathbf{a}_2=\bmatrix 2\\ 3 \endbmatrix$ , and write the result in the form $A=Q\cdot R$ .
(8 points) Let
- Find the characteristic polynomial of A.
- Find the eigenvalues of A.
- Find a basis for each eigenspace of A.
- Find a diagonal matrix $\Lambda$ and an invertible matrix S such that $A=S\cdot\Lambda\cdot S^{-1}$ .
  [You do not have to calculate S^-1.]
(8 points) Let A be a 3 by 3 matrix, with eigenvalues , , .
- Compute $\tr(A)$ and $\det(A)$ .
- Is A invertible? Explain your answer.
- Is A diagonalizable? Explain your answer.
- Compute $\tr(A^3)$ and $\det(A^3)$ .

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Created by Alexandru Suciu, Mon Nov 24, 1997

alexsuciu@neu.edu

http://www.math.neu.edu/~suciu/mth1230/1230f97.q4/index.html