**Readings Questions:**

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 aAs before vectors are linearly dependent if they are not independent._{1}, a_{2}, ... , a_{n }are linearly independent if and only if c_{1}a_{1}+ c_{2}a_{2}+ ... ++ c_{n}a_{n}= 0 implies c_{1}= c_{2}= ... = c_{n}= 0.

Thus the set of vectors aProve 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._{1}, a_{2}, ... , a_{n }are linearly dependent if and only if

c_{1}a_{1}+ c_{2}a_{2}+ ... ++ c_{n}a_{n}= 0 with at least one c_{i}not equal to zero. (*)

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

a_{1}**i**+a_{2}**j**+a_{3}**k
**=
(a_{1},a_{2},a_{3}).

**Problems:**

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