How to find the basis of a vector space V?

How to find the basis of a vector space?

How to find the basis of a vector space V? In order to find the basis of a vector space , we need to check two properties:

  1. The vectors should be linearly independent.
  2. These vectors should span in that vector space.

If both of these properties hold, then it means the given set of vectors form the basis otherwise not.

What are the standard basis of R2?

The standard basis of R2 is composed of two vectors v1(1,0) and v2=(0,1). The reason is, these two vectors can be used for generating any vector having two components. These two vectors are actually the axes : x-axis and y-axis. The first component of v1 indicates the presence of x axis while y-axis is missing. Similarly in v2, the second component=1 indicating the presence of y axis and all x-component along it is equal to zero.

basis of R2
basis of R2

Lets say we want to see first property i.e., v1 and v2 are linearly independent or not. For this solve

c1v1+c2v2=0

c1(1,0)+c2(0,1)=(0,0)

It can be seen easily only c1=0 and c2=0. No other values of c1 and c2 satisfy the above equation. That is why v1 and v2 are linearly independent.

Now, check either they span the vector space R2 or not?

For this we will try to generate any vector having components (a1,a2) using the linear combinations of these two vectors.

(a1,a2)=c1v1+c2v2

Lets say

a1=5, a2=6 so we will have

(5,6)=c1v1+c2v2

(5,6)=c1(1,0)+c2(0,1)

1.c1+0.c2=5

0.c1+1.c2=6

By solving these equations, we will get c1=5 while c2=6

As

(5,6)=5(1,0)+6(0.1)

So, it is clear that we can generate any vector in R2 using these vectors. So v1 and v2 span the vector space V.

From the above explanation both properties of the basis are satisfied, that is why v1 and v2 form the basis in R2

Also read here 

what is the vector space in linear algebra? vector space example

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