Linear Algebra - 1016 - 432
Fall 2006
Set 3
Due: Wednesday, October 25, at the start of class
Please write up the following problems
a. section 5.2 , problems 12 , 18 , 20 , 26 , 30
b. section 5.3 , problem 24
Note: Use our tidy way of characterizing the column space of a matrix. For given matrix M, and vector v,
we showed that
€
v∈col M
( )
if and only if
€
Mw =v
, for some vector w.
c. section 7.3 , problems 16 , 48
Note: In problem 16, follow the simple method we used in class. If you wish, you may use a calculuator,
or a computer algebra system like Maple, to help with some of the intermediate matrix
computations. However, you should still show the steps needed to get the result.
d. section 6.1 , problems 4 , 36 , 62
e. section 6.2 , problems 8 , 14 , 28 , 36 , 48 , 54
f. Recall from class that the subsets of a given set S form a vector space over
€
Z2
. Operations were as follows:
i.
€
H⊕K=H∪K
( )
−H∩K
( )
, for subsets H and K of S
ii.
€
0⋅H=∅
and
€
1⋅H=H
Let
€
S=1, 2 , 3, 4
{ }
and V be the vector space of subsets of S, given in the above way. Let W be the
subspace of V spanned by the 2-element subsets of S.
•Determine the dimension of W.
•What else is in W, apart from the spanning set?
•What sizes may may occur for other subspaces of V?