Prof. A. Suciu

MTH 1230 -- Linear Algebra -- Fall 1997

Quiz 3

(13 points) Let .
- 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 matrix. Suppose that , 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.
$\begin{displaymath} \vec v_1=\bmatrix 2\\ 2\\ 6\endbmatrix \qquad \vec v_2=\bmat... ...\ 5\endbmatrix \qquad \vec v_3=\bmatrix -5\\ 7\\ -3\endbmatrix \end{displaymath}$
(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