MAT 2401 Discovery Lab 4.1
Objectives
 To explore the vectors in the plane.
 To investigate the 2 important operations: vector addition and scalar multiplication.
 To explore the definition and structure of real vector spaces.
Instructions
 Do not look up any references including the textbook and internet.
 Use correct notations and do not skip steps.
 Two persons per group.
 Do not communicate with other groups.
Vectors in the Plane
A vector in the plane is represented graphically by a
arrow from the origin to the point  x1 , x2  .
We will abuse the notation and represent the vector x
by the same ordered pair , i.e. x   x1 , x2  .
Vector Addition
Let u   u1 , u2  and v   v1 , v2  , the sum of the two vectors is defined as
u  v  u1 , u2    v1 , v2   u1  v1 , u2  v2 
Scalar Multiplication
Let u   u1 , u2  and c a scalar, then the vector cu is defined as
cu  c u1 , u2    cu1 , cu2 
Remark
One can define the vector subtraction in terms of vector addition u  v  u   1 v .
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1. Let u   5, 4  and v   3, 2  .
(a) Compute u  v and 2v .
(b) Draw u , v , u  v , and 2v on the graph paper. Make sure you label the vectors.
uv 
2v 
2. Let u   4,6 and v   2,3 .
(a) Compute
1
uv.
2
(b) Draw u , v , and
1
u  v on the graph paper. Make sure you label the vectors.
2
1
uv 
2
2
Properties
3. (a) What is the zero vector in properties 4 and 5? Write it in the component form
u   u1 , u2  .

0


,




(b) What is the identity in property 10? Is it a vector or a scalar?
(c) How do you know properties 1 and 6 are true? Explain.
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(d) Use the definitions of the two standard operations to show that property 7 is true.
Partial solutions are given. Fill in the missing steps.
Let u   u1 , u2  and v   v1 , v2  ,
cu  cv  c  u1 , u2   c  v1 , v2 








 

 
 
,




,
 

 
 
Multiply c into each
vector.
Add the 2 vectors.








 
,c
 
 
,




,
,
 
 c
 
 

 c



 c 

Hints
factor out c from each
component.
Factor out c from the
vector.




,
 c u  v 
Break up the vector into
2 vectors.
(e) Use the definitions of the two standard operations to show that property 8 is true.
Partial solutions are given. Fill in the missing steps.
Let u   u1 , u2  .
Hints
 c  d  u   c  d  u1 , u2 













u2 


u1 ,
u1 

 c
,


 cu  du
,
u1 ,

u2 


u2 
 

 
 


,
d




Multiply c  d into the
vector.
,








Break up the vector into
2 vectors.
Factor out c and d
from the vectors.
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It is easy to see that we can generalize the notion of vectors to higher dimensions in a
similar fashion.
n Dimensional Real Vector Space Rn
Denote the n-space as
R n   x1 , x2 ,
, xn  | x1 , x2 ,
, xn  R
with the standard operations.
Vector Addition
Let u   u1 , u2 , , un  and v   v1 , v2 ,
u  v   u1 , u2 ,
, vn  , the sum of the two vectors is defined as
, un    v1 , v2 ,
, vn   u1  v1 , u2  v2 ,
, un  vn 
Scalar Multiplication
Let u   u1 , u2 , , un  and c a scalar, then the vector cu is defined as
cu  c  u1 , u2 ,
, un    cu1, cu2 ,
, cun 
Similarly, we have the following properties
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Space of Polynomial of Degree At Most 2
Let P2 be the collection of polynomials of degree at most 2, i.e.


P2  a0  a1 x  a2 x 2 | a0 , a1 , a2  R
4. (a) Suppose we want P2 to have the kind of 10 properties as shown above. How
would you define the polynomial addition and scalar multiplication?
Polynomial Addition
Let p  x   a0  a1 x  a2 x 2 and q  x   b0  b1 x  b2 x2 , the sum of the two polynomials is
defined as

 

p  x   q  x   a0  a1 x  a2 x 2  b0  b1 x  b2 x 2 

x
x2
Scalar Multiplication
Let p  x   a0  a1 x  a2 x 2 and c a scalar, then the polynomial cp  x  is defined as


cp  x   c a0  a1 x  a2 x 2 

x
x2
(b) What is the zero polynomial in properties 4 and 5? Write it in the form of
a0  a1 x  a2 x 2 .
0  0 x 

x
x2
(b) What is the identity in property 10? Is it a polynomial or a scalar?
(c) How do you know properties 1 and 6 are true? Explain.
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Let Let P2 be the collection of polynomials of degree 2, i.e.


P2  a0  a1 x  a2 x 2 | a0 , a1 , a2  R, a2  0
5. Use the standard operations defined in problem 4, do you think P2 has all of the 10
properties as shown above? Explain your precise reasons.
Let V  R 2 together with the vector addition defined above in p.1 and a non-standard
scalar multiplication given by
cu  c  u1 , u2    cu1 ,0
6. Do you think V has all of the10 properties as shown above? Explain your precise
reasons.
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