10.4: Review Problems
( \newcommand{\kernel}{\mathrm{null}\,}\)
1. Let Bn be the space of n×1 bit-valued matrices (i.e., column vectors) over the field Z2.
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.
a) How many different vectors are there in Bn?
b) Find a collection S of vectors that span B3 and are linearly independent. In other words, find a basis of B3.
c) Write each other vector in B3 as a linear combination of the vectors in the set S that you chose.
d) Would it be possible to span B3 with only two vectors?
2. Let ei be the vector in ℜn with a 1 in the ith position and 0's in every other position. Let v be an arbitrary vector in ℜn.
a) Show that the collection {e1,…,en} is linearly independent.
b) Demonstrate that v=∑ni=1(v⋅ei)ei.
c) The span{e1,…,en} is the same as what vector space?
3. Consider the ordered set of vectors from ℜ3
$$
\left( (123)
\]
a) Determine if the set is linearly independent by using the vectors as the columns of a matrix M and finding RREF(M).
b) If possible, write each vector as a linear combination of the preceding ones.
c) Remove the vectors which can be expressed as linear combinations of the preceding vectors to form a linearly independent ordered set. (Every vector in your set set should be from the given set.)
4. Gaussian elimination is a useful tool figure out whether a set of vectors spans a vector space and if they are linearly independent.
Consider a matrix M made from an ordered set of column vectors (v1,v2,…,vm)⊂Rn and the three cases listed below:
a) RREF(M) is the identity matrix.
b) RREF(M) has a row of zeros.
c) Neither case i or ii apply.
First give an explicit example for each case, state whether the column vectors you use are linearly independent or spanning in each case. Then, in general, determine whether (v1,v2,…,vm) are linearly independent and/or spanning Rn in each of the three cases. If they are linearly dependent, does RREF(M) tell you which vectors could be removed to yield an independent set of vectors?
Contributor
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)