# 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 .
• Find .
• Find .
• Find .
• Is ?     YES      NO     MAYBE
• Is ?     YES      NO     MAYBE
• Is ?     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.

• (12 points) In each of the following, a vector space V and a subset S are given. Circle one answer:
• ,
 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

• ,
 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

• ,
 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

• ,
 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