MAT 242 Dr. Ibrahim
SLN: 80071 & 80072 Fall, 2007
MAT 242 Assignment C
C1: Which of the following subsets of IR2are subspaces. Briefly justify your answer.
a) [x, y] satisfying x= 2y;
hAnswer: Yes, this is a vector subspace. It’s a line through the origin in IR2, in the
direction of the vector (2,1)T.i
b) [x, y] satisfying x= 2yand y= 2x;
hAnswer: This is a vector subspace of IR2consisting of the (0,0) vector only.i
c) [x, y] satisfying x= 2y+ 1;
hAnswer: Not a vector space, because it is not closed under multiplication: 2(2,1) =
(4,2).i
d) [x, y] satisfying xy = 0;
hAnswer: Not a vector space, because it is not closed under addition: (1,0) + (0,1) =
(1,1).i
e) [x, y] satisfying x≥0 and y≥0.
hAnswer: Not a vector space, because it is not closed under multiplication: −(2,1) =
(−2,−1).i
C2: For which real numbers λare the following vectors linearly independent in IR3?
X1=
λ
−1
−1
, X2=
−1
λ
−1
, X3=
−1
−1
λ
.
hAnswer: The three vectors are linearly dependent only if λ=−1 or λ= 2.i
C3: Compute the reduced row echelon form Rof the follwoing matrix Aand find bases for
its row, column and null spaces of A.
A=
1 1 2 0 1 3
2 1 4 0 3 2
0 0 0 1 3 0
3 0 2 4 3 2
.
