PROF. A. SUCIU MTH 1230 - LINEAR ALGEBRA FALL 1997

SAMPLE QUIZ 4

• Find the least squares solution of the inconsistent system $A\mathbf{x} = \mathbf{b}$ for

  $$A=\bmatrix 4&0\\ 0&2\\ 1&1 \endbmatrix,\qquad \mathbf{b}= \bmatrix 2\\ 0\\ 11\endbmatrix.$$

  
  

• Find the equation of the least-squares line that fits the following data points:

  $$\begin{array} {\vert c\vert\vert c\vert c\vert c\vert c\vert} \hline x&1&2&4&5\\ \hline y&0&1&2&3\\ \hline\end{array}$$

  Sketch the resulting line. What is the predicted value of y at x=6, based on this model?

  
  

• Let      $${ \mathbf{a}_1=\bmatrix 1\\ 1\\ 1\\ 1 \endbmatrix , \quad\quad \ma... ...-1 \endbmatrix , \quad\quad \mathbf{a}_3=\bmatrix 1\\ 1\\ 1\\ 0 \endbmatrix . }$$
  •  Find an orthonormal basis for the subspace of $\mathbb{R}^4$ which has $\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$ as a basis.
  •  Use part one to find the QR factorization of the matrix $A=\bmatrix \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \endbmatrix$.
  
  
• Let

  $$A=\bmatrix 4&-7&0\\ 2&-5&0\\ 0&0&2 \endbmatrix.$$

  • 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.]
  
  
• A $2\times 2$ matrix A has first row $\bmatrix -2 & 5 \endbmatrix$ and eigenvalues $\lambda_1=-1$ and $\lambda_2=3$.
  • Find A.
  • What are the eigenvalues of A^-1?
  • Compute $\det ( A^{-1}+I)$, where I is the identity $2\times 2$ matrix. Explain your result.
  
  
• A $4 \times 4$ matrix has eigenvalues $\lambda_1=1, \lambda_2=2, \lambda_3=3, \lambda_4=4$.
  • Find the eigenvalues of A².
  • Find the trace of A².
  • Find the determinant of A².