Linear algebra/Basis for a vector space

The definition of a basis for a vector space will be given first. This definition is not always the most useful one tool for proving that a subset of a vector space forms a basis for that space. Therefore, we will look at some theorems that can be developed from this fundamental definition, in this section and the following one [[Dimension of a Vector Space]], that provide easier criteria for determining whether a subset of a vector space forms a basis for that vector space.

Definition I: Let {v₁,v₂,…,v_n} be a subset of an arbitrary vector space V. If these elements both generate V and are linearly independent they form a basis for V.

Example I: Show that the vectors (1,1) and (-1,2) form a basis for R².

Proof: We have to prove that these 2 vectors are both linearly independent and that they generate R².

Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that:

• a(1,1)+b(-1,2)=(0,0)
  

Then:

• (a-b,a+2b)=(0,0) and a-b=0 and a+2b=0.
  

Subtracting the first equation from the second, we obtain:

• 3b=0 so b=0.
  

And from the first equation then:

• a=0.
  

Part II: To prove that these two vectors generate R², we have to let (a,b) be an arbitrary element of R², and show that there exist numbers x,y such that:

• x(1,1)+y(-1,2)=(a,b)
  

Then we have to solve the equations:

• x-y=a
• x+2y=b.
  

Subtracting the first equation from the second, we get:

• 3y=b-a, and then
• y=(b-a)/3, and finally
• x=y+a=((b-a)/3)+a.
  

Example II: We have already shown that E₁, E₂,…,E_n are linearly independent and generate Rⁿ. Therefore, they form a basis for Rⁿ.

Example III: Let W be the vector space generated by e^t, e^2t. We have already shown they are linearly independent. Then they form a basis for W.