Name:
















Instructor:

NORTHEASTERN UNIVERSITY
Department of Mathematics

MTH 1230 FINAL EXAM Fall 1997
Instructions: Put your name and the instructor's name in the blanks above. Put your final answers to each question in the designated spaces--you may lose credit if you don't. Calculators are permitted. A single cribsheet of formulas is allowed. SHOW YOUR WORK. If there is not enough room to show your work, use the back of the preceding page. Good luck!

• 8 points Are the following vectors independent or dependent? If they are independent, say why. If they are dependent, exhibit a linear dependence relation among them.

  $$\mathbf{v}_1=\bmatrix 3\\ 0\\ -1\endbmatrix \qquad \mathbf{v... ...endbmatrix \qquad \mathbf{v}_3=\bmatrix 1\\ -4\\ -7\endbmatrix$$

  
  
  

• 20 points Let $A=\bmatrix 2 & 0 & 3 \\ 4 & 5 & 7 \\ -6 & 10 & -7 \endbmatrix$,    $b=\bmatrix -2\\ -3 \\ 8 \endbmatrix$.
  • Solve the system of linear equations Ax=b, indicating clearly the row operations, pivots, leading variables, and free variables.
    
    
  • Find the LU-decomposition of the matrix A.
    
  • Find a basis for each of the row space, column space, and null-space of A.
    
    
  • Find the rank of A.
    
  • Find a non-zero vector that is orthogonal to both the vectors $\bmatrix 2 & 0 & 3\endbmatrix$ and $\bmatrix -6 & 10 & -7 \endbmatrix$.
    

• 12 points In each of the following, a vector space V and a subset S are given. Circle one answer:
  
  • $V=\mathbb{R}^4$, $S=\{(-t,4t,3t,0)\mid t \text{ is a real number}\}$
    

    S is closed under addition:     YES NO

    S is closed under scalar multiplication:     YES NO

    S is a vector subspace of V:     YES NO

    
    
    
    
  • $V=\mathbb{R}^4$, $S=\{(-t,4t,3t,1)\mid t \text{ is a positive real number}\}$
    

    S is closed under addition:     YES NO

    S is closed under scalar multiplication:     YES NO

    S is a vector subspace of V:     YES NO

    
    
    
    
  • $V=\mathbb{R}^4$, $S=\{x\in V\mid x_1+2x_2-x_3=0,\quad 2x_1+x_3-3x_4=0\}$
    

    S is closed under addition:     YES NO

    S is closed under scalar multiplication:     YES NO

    S is a vector subspace of V:     YES NO

    
    
    
    
  • $V=\mathbb{R}^4$, $S=\{x\in V\mid x_1+2x_2-x_3\ge 0\}$
    

    S is closed under addition:     YES NO

    S is closed under scalar multiplication:     YES NO

    S is a vector subspace of V:     YES NO

    
    

• 12 pts
  • Find the $3\times 3$ matrix A associated with the linear mapping that rotates the yz-plane by $-60^{\circ}$ and reflects the x-axis about the yz-plane.
    
    
  • Is A orthogonal?
    
  • What is $\det(A)$?
    
  • What is the image of the point (-3,2,1) under the above mapping?
    

• 12 pts Let $A=\bmatrix 1&2\\ 0&1\\ 1&0\\ 1&1\endbmatrix$ and $b=\bmatrix -2\\ 3\\ 0\\ 4\endbmatrix$.
  •  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.
    

• 16 pts Let    $\mathbf{a}_1=\bmatrix 3\\ 4\endbmatrix , \quad \mathbf{a}_2=\bmatrix -1\\ 1\endbmatrix .$
  • Find unit vectors in the direction of $\mathbf{a}_1$ and $\mathbf{a}_2$, respectively.
    
  • Find the lengths of $\mathbf{a}_1$ and $\mathbf{a}_2$, and compute the dot product $\mathbf{a}_1\cdot\mathbf{a}_2$.
    
  • Find the angle between $\mathbf{a}_1$ and $\mathbf{a}_2$. Are $\mathbf{a}_1$ and $\mathbf{a}_2$ orthogonal?
    
  • Let $A=\bmatrix \mathbf{a}_1 & \mathbf{a}_2\endbmatrix$. Use the Gram-Schmidt process to find the QR-factorization of A.
    
    

• 20 pts Let $A=\bmatrix 2&2\\ 3&1\endbmatrix.$
  • Find the characteristic equation for A.
    
  • Find the eigenvalues of A.
    
  •   Find a basis for each eigenspace of A.
    
    
  • Form a matrix S using the two independent eigenvectors from part c as column vectors, and calculate S^-1.
    
  • Calculate $S^{-1}\cdot A\cdot S$. Explain your answer.