Unit 4 - Activity 8

ACTIVITY 4.8
INTRODUCTION TO THE BASES - THE CANONICAL BASE

In the activity 2.10 we have seen what it means to express a vector like a linear combination of two vectors and , and to find two scalars x and y that verify = x+ y.

In this activity we will study what it happens with vectors and when we choose vectors (1,0) and (0,1). These two vectors become accustomed to identify with two concrete letters, that they are and , therefore, we will put (1,0)= and (0,1)=.

Put a vector =(u₁,u₂) like a linear combination of the two vectors and will mean to find two scalars x and y that they verify= x+ y. This is equivalent to (u₁,u₂)=x(1,0)+y(0,1), and it is easy to get x and y: there is enough with making x=u₁ and y=u₂. That is to say, any vector=(u₁,u₂) can put as a linear combination of and , writing (u₁,u₂) = u₁ + u₂.

Vectors=(1,0) and =(0,1) call canonical base of vectors of the plane, and the scalars u₁ and u₂, components of=(u₁,u₂) in the canonical base.

Surely you will ask yourself about how are the "habitual" components of a vector, now we will call them components of the canonical base. Be patient: the importance of all that we have explained you, you will see it in the next activity, where you will see that it can be bases not canonical and components of a vector in a non canonical base.

INTERACTIVE ACTIVITY

a) Draw the next given vectors such as linear combinations of vectors and of the canonical base:

= 7+ 5

= 6

= -+ 2

= - 5

= - 4 - 3

= 4

Which components do they have canonical base?

b) Draw the next given vectors by its components in the base and :

= (4 , 6)

= (6 , 0)

= (-2 , 1)

= (0 , -3)

= (-3 , -4)

= (0 , 4)

Which expression does it have like linear combination of the vectors with canonical base?

SOLUTION