Linear algebra/Basis for a vector space

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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,…,vn} 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,…,En are linearly independent and generate Rn. Therefore, they form a basis for Rn.

Example III: Let W be the vector space generated by et, e2t. We have already shown they are linearly independent. Then they form a basis for W.