Linear Algebra 1 (MA 371), Spring, 2000—2001
Quiz 4 (Thursday, March 29, 2001)
NAME:
BOX:





1
2 0 −1
1 0 0
1 2 0 −1




1) (10 pts) Given that A =  2
4 1 0  =  2 1 0  0 0 1
2 
 = LU,
−1 −2 2 5
−1 2 1
0 0 0
0
A : R −→ R
Fill in the boxes
dim(row space of A) =
dim(N (A)) =
.
dim(R(A)) =
Basis for N (A) is
Basis for the row space of A is
Basis for R(A) is
Basis for R(U) is
Why is N (L) =
−
→
0 ?
2) (3 pts) Suppose that U is obtained from A by elementary row operations. Prove that
N (A) = N (U ). This is for a general m × n matrix A, not the matrices in (1).
3) (13 pts) Suppose that A is m × n and A = LU is the LU decomposition of A. In the space
to the left of the statement.put T if the statement is always true; otherwise put F.
(a) L singular.
(b) N (A) = N (U ).
(c) R(A) = R(U ).
(d) Row space of A = row space of U.
(e) dim(R(A)) = dim(R(U ))
(f) rank(A) = m.
(g) N (A) is a subspace of Rm
(h) dim(R(A))+ dim(N (A)) = m.
(i) If R(A) = Rm , then Ax = b is consistent for every b.
(j) The span of the columns of A is equal to the span of the columns of U.
(k) The columns of A form a basis for R(A).
(l) The non—zero rows of U form a basis for the row space of A.
(m) If m = n and R(A) = Rm , then A is invertible.



 


4
−3 
 2




 

4) (3 pts) Is  5  ∈ span  1  ,  2 ? Show your work.

2
0
1 
5) (3 pts) What is the definition of a basis for a vector space V ? Start your answer with “A
set of vectors B = {v1 , v2 , ..., vn } is a basis for V if ”
6)
Use 
thedefinition
of linear independence to show that the set of vectors
(3 pts)
 



1
0
1



 
 

 0  ,  1  ,  0  is linearly independent in R3 .



0
1
1 