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

QUIZ 2

• •  Find the LU-decomposition of    $A=\bmatrix 3 & 0 & 1 \\ -6 & 4 & -3 \\ 3 & 2 & 1 \endbmatrix.$
    
  • Use the LU-decomposition of A from part (a) to solve    $A\cdot \bmatrix x_1\\ x_2\\ x_3 \endbmatrix = \bmatrix 3\\ 4\\ 2 \endbmatrix.$
  
  
• Use Gaussian elimination to find the inverse of the matrix    $A=\bmatrix 1&0&1\\ 2&2&0\\ 1&1&1\endbmatrix.$

  
  

• Consider the matrix    $A=\bmatrix 1&2\\ 3&4\endbmatrix.$
  •   Find elementary matrices E₁, E₂, and E₃ such that $E_3 \cdot E_2 \cdot E_1 \cdot A = I$.
    
  •   Use part (a) to express A as a product of elementary matrices.
    
  •   Use part (a) to express A^-1 as a product of elementary matrices.
    
  • Use part (c) to find A^-1.
  
  
• Let    $\vec a=\bmatrix 1&1&-1\endbmatrix , \quad \vec b=\bmatrix 2&1&2\endbmatrix .$
  • Find unit vectors in the direction of $\vec a$ and $\vec b$, respectively.
    
  • Find the lengths of $\vec a$ and $\vec b$, and compute the dot product $\vec a \cdot \vec b$.
    
  • Find the angle between $\vec a$ and $\vec b$. Are $\vec a$ and $\vec b$ orthogonal?
    
  • Find the projection of $\vec a$ onto $\vec b$.
    
  • Find the projection of $\vec b$ onto $\vec a$.