Assignment For Section 1.5

You should read the section before attempting any reading questions or problems. Remember that you are responsible for all definitions and examples. Also remember to keep the answers to the reading questions on a separate piece of paper; these will be randomly collected from time to time.

1)   Verify that if a  and  b  are placed tail to tail than b - a  is the vector that points from the head of  a  to the head of  b.  Hint:  Start with  a  and  b  and then consider  (-1)a  and b+(-1)a.

2)   In example 1.5.2 explain why is it sufficient to show  (ie  explain why  is parallel to ).

3)    Corollary 1.5.1 can be used to extend the definition of linearly independent to several vectors:

The set of vectors a1, a2, ... , an  are linearly independent if and only if c1a1+ c2a2+ ... ++ cnan= 0  implies c1 = c2 = ... = cn= 0.
As before vectors are linearly dependent if they are not independent.
Thus the set of vectors a1, a2, ... , a are linearly dependent if and only if
c1a1+ c2a2+ ... ++ cnan = 0 with at least one ci not equal to zero.  (*)
Prove that this extended definition is the same as the definition given in problem 30 (pg 48).  This will involve showing two things:  If one vector can be written as a linear combination of the others then (*) holds true and if  (*) is true then one of the vectors can be written as a linear combination of the others.  Note that linear combination is defined in problem 27.

4)   Consider example 1.5.3.
a)    Explain why   and  are linearly dependent.
b)    Explain why .

5)   Use the definition of scalar multiplication, 1.5.3, and vector addition, 1.5.2, to prove that
a1i+a2j+a3k = (a1,a2,a3).

Problems:

page 44     1, 3, 5, 7, 8a,b,  10a, 11, 13, 15, 16, 19, 21, 25, 27, 30