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Ordered bases

Let V be a vector space over the field F having dimension n.

Let {v1,,vn} be a basis for V. We call (v1,,vn) an ordered basis for V.

By fixing an order on the basis elements, we can now represent every vector in V as an n-tuple.

More precisely, let vV. Recall that there exist scalars λ1,,λnF such that v=λ1v1++λnvn.

Define [v](v1,,vn) to be the n-tuple [λ1λ2λn].

We call [v](v1,,vn) a tuple representation (or coordinate representation of v with respect to the ordered basis (v1,,vn).

Clearly, if we choose a different basis or a different ordering of the basis vectors, we get a different representation for the same vector. That is why it is so important to specify the ordered basis when working with tuple representations.

Examples

  1. Note that Γ=([11],[23]) is an ordered basis for R2. Let u=[32]. What is [u]Γ?

    Solution. We first write u as a linear combination of the ordered basis elements. That is, we want to find real numbers α and β such that u=α[11]+β[23]. Thus, we need to solve the system 3=α+2β2=α+3β Solving gives α=1 and β=1. Hence, [u]Γ=[11].

  2. Let V denote the vector space of polynomials in x with real coefficients having degree at most 2. Let Γ denote the ordered basis (1,x+1,x2+1) and let Ω denote the ordered basis (x1,x2+1,x21). (Check that these are indeed ordered bases.) We want to find the tuple representations of x with respect to these ordered bases.

    To determine [x]Γ, we need to find real numbers α,β,γ such that x=α(1)+β(x+1)+γ(x2+1). Simplifying the right-hand side, we obtain x=γx2+βx+(α+β+γ). Since the left-hand side has no x2 term, we must have γ=0. Now, comparing the coefficients of x on both sides, we get β=1. Since the left-hand side has no constant term, we must have α=1.

    In other words, x=(1)1+1(x+1)+0x2, giving [x]Γ=[110].

    For [x]Ω, note that x=1(x1)+12(x2+1)+(12)(x2). Hence, [x]Ω=[11212].

Working with tuple representations

The beauty of these tuple representations of vectors is that we can now work with tuples instead. For instance, if u,vV and Γ is an ordered basis for V, then one can easily check that [u+v]Γ=[u]Γ+[v]Γ. In other words, adding two vectors in V simply requires adding the corresponding tuple representations in Fn.

And if λ is a scalar, then [λu]Γ=λ[u]Γ. Thus, multiplying a vector by a scalar simply involves multiplying the corresponding tuple representation by the same scalar.

Quick Quiz

Exercises

  1. Let Γ=([12],[23]) and Ω=([11],[12]) be two ordered basis for R2.
    1. Let u=[10] and v=[01]. Find the following:

      1. [u]Γ  

      2. [u]Ω  

      3. [v]Γ  

      4. [v]Ω  

    2. Give a 2×2 matrix A such that [x]Ω=A[x]Γ for all xR2.