Page 210 - 35Linear Algebra
P. 210
210 Linear Independence
10.4 Review Problems
Reading Problems 1 ,2
Testing for linear independence 3, 4
Webwork:
Gaussian elimination 5
Spanning and linear independence 6
n
1. Let B be the space of n × 1 bit-valued matrices (i.e., column vectors)
over the field Z 2 . Remember that this means that the coefficients in
any linear combination can be only 0 or 1, with rules for adding and
multiplying coefficients given here.
n
(a) How many different vectors are there in B ?
3
(b) Find a collection S of vectors that span B and are linearly inde-
3
pendent. In other words, find a basis of B .
3
(c) Write each other vector in B as a linear combination of the vectors
in the set S that you chose.
3
(d) Would it be possible to span B with only two vectors?
Hint
n
2. Let e i be the vector in R with a 1 in the ith position and 0’s in every
n
other position. Let v be an arbitrary vector in R .
(a) Show that the collection {e 1 , . . . , e n } is linearly independent.
P n
(b) Demonstrate that v = (v e i )e i .
i=1
(c) The span{e 1 , . . . , e n } is the same as what vector space?
3
3. Consider the ordered set of vectors from R
1 2 1 1
2 , 4 , 0 , 4
3 6 1 5
(a) Determine if the set is linearly independent by using the vectors
as the columns of a matrix M and finding RREF(M).
210
