Prof. A. Suciu MTH 1230 -- LINEAR ALGEBRA Fall 1997

QUIZ 3

• 13 points Let $A=\bmatrix 2&1&1&0\\ 4&2&2&0\\ -6&-3&0&1\\ 2&1&-2&0\endbmatrix$.
  • Find a basis for the row space of A.
    
  • Find a basis for the column space of A.
    
  • Find a basis for the null space of A.
  
  
  
• 7 points Let A be a $12\times 20$ matrix. Suppose that $A=L\cdot U$, where L is a lower triangular matrix, with 1's down the diagonal, and U is a row-echelon matrix, with 9 non-zero pivots.
  
  • Find $\rank A$.
    
  • Find $\dim \RS(A)$.
    
  • Find $\dim \CS(A)$.
    
  • Find $\dim \NS(A)$.
    
  • Is $\RS(A)=\RS(U)$?     YES     NO     MAYBE
    
  • Is $\CS(A)=\CS(U)$?     YES     NO     MAYBE
    
  • Is $\NS(A)=\NS(U)$?     YES     NO     MAYBE
    

• 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.

  $$\vec v_1=\bmatrix 2\\ 2\\ 6\endbmatrix \qquad \vec v_2=\bmat... ...\ 5\endbmatrix \qquad \vec v_3=\bmatrix -5\\ 7\\ -3\endbmatrix$$

  
  
  
• 12 points In each of the following, a vector space V and a subset S are given. Circle one answer:
  
  • $V=\mathbb{R}^3$, $S=\{(t,2t,3t)\mid t \text{ is a real number}\}$
    

    S is closed under addition: YES NO MAYBE

    S is closed under scalar multiplication: YES NO MAYBE

    S is a vector subspace of V: YES NO MAYBE

    
    
  • $V=\mathbb{R}^3$, $S=\{(t,2t,3t)\mid t \text{ is a positive real number}\}$
    

    S is closed under addition: YES NO MAYBE

    S is closed under scalar multiplication: YES NO MAYBE

    S is a vector subspace of V: YES NO MAYBE

    
    
  • $V=\mathbb{R}^3$, $S=\{(t,2t,3t)\mid t \text{ is an integer}\}$
    

    S is closed under addition: YES NO MAYBE

    S is closed under scalar multiplication: YES NO MAYBE

    S is a vector subspace of V: YES NO MAYBE

    
    
  • $V=\mathbb{R}^3$, $S=\{(t+1,2t,3t-1)\mid t \text{ is a real number}\}$
    

    S is closed under addition: YES NO MAYBE

    S is closed under scalar multiplication: YES NO MAYBE

    S is a vector subspace of V: YES NO MAYBE