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

SAMPLE QUIZ 3

• Let    $A=\bmatrix 1&2&3&4\\ 0&1&1&3 \\ 0&1&1&4 \endbmatrix.$Find:
  • A basis for the row space of A.
  • A basis for the column space of A.
  • A basis for the null space of A.
  • The rank of A.
  
  
• In each of parts first and second, are the given vectors $\vec v_1, \vec v_2, \vec v_3$ independent or dependent? If they are independent, say why. If they are dependent, exhibit a linear dependence relation among them.
  •       $\vec v_1=\bmatrix 1\\ 0\\ 0\endbmatrix \qquad \vec v_2=\bmatrix 2\\ 1\\ -3\endbmatrix \qquad \vec v_3=\bmatrix -1\\ 2\\ -6\endbmatrix$
  •       $\vec v_1=\bmatrix 1\\ 0\\ 0\endbmatrix \qquad \vec v_2=\bmatrix 2\\ 1\\ -3\endbmatrix \qquad \vec v_3=\bmatrix -1\\ 2\\ 6\endbmatrix$
  
  
• Which of the given subsets of the vector space, M₂₃, of all $2\times 3$ matrices are subspaces? In each case, give a reason for your answer.
  
  • All matrices of the form    $\bmatrix a&b&c\\ d&0&0\endbmatrix$,    where b=a+c.
  • All matrices of the form    $\bmatrix a&b&c\\ d&0&0\endbmatrix$,    where c>0.
  • All matrices of the form    $\bmatrix a&b&c\\ d&e&f\endbmatrix$,    where a=-2c and f=e+2d.
  • All matrices of the form    $\bmatrix a&b&c\\ d&e&f\endbmatrix$,    where a=2c+1.
  
  
• Which of the following sets of vectors span $\mathbb{R}^4$? In each case, give a reason for your answer.
  
  • $\vec v_1=\bmatrix 1&2&5&7\endbmatrix$, $\vec v_2=\bmatrix 1&1&-4&0\endbmatrix$, $\vec v_3=\bmatrix 0&0&0&3\endbmatrix$.
    
  • $\vec v_1=\bmatrix 1&0&0&1\endbmatrix$,$\vec v_2=\bmatrix 0&1&0&0\endbmatrix$,$\vec v_3=\bmatrix 1&1&1&1\endbmatrix$,$\vec v_4=\bmatrix 1&1&1&0\endbmatrix$.
    
  • $\vec v_1=\bmatrix 1&1&0&0\endbmatrix$,$\vec v_2=\bmatrix 0&0&1&1\endbmatrix$,$\vec v_3=\bmatrix 1&1&1&1\endbmatrix$,$\vec v_4=\bmatrix 2&2&-1&-1\endbmatrix$.
    
  • $\vec v_1=\bmatrix 3&2&1&0\endbmatrix$, $\vec v_2=\bmatrix 1&0&0&0\endbmatrix$, $\vec v_3=\bmatrix 0&1&7&1\endbmatrix$, $\vec v_4=\bmatrix 0&0&1&2\endbmatrix$, $\vec v_5=\bmatrix 0&6&1&0\endbmatrix$.