User:IssaRice/Linear algebra/Change of basis example in two dimensions

From Machinelearning

This example comes from this video. To make it easier to go back and forth between this page and the video, the notation on this page follows that of the video.

We are working in R2, the plane.

Jennifer's basis vectors: b→1:=[21] and b→2:=[−11]

To Jennifer, b→1 looks like [10] and b→2 looks like [01].

If Jennifer says "[−12]", to us (in the standard basis) this is the vector −1b→1+2b→2=−1[21]+2[−11]=[−41].

We can also write the above calculation as [b→1b→2][−12]=[2−111][−12]=[−41].

Notice that [b→1b→2]e→1=b→1 and [b→1b→2]e→2=b→2, i.e., this matrix transforms our (standard) basis vectors into Jennifer's basis vectors.

How can we write this using change of basis notation? When Jennifer says "[−12]", this is the vector v∈R2 such that, when written in Jennifer's coordinate system, it has coordinates (−1,2). In other words, it is the vector v such that [v](b→1,b→2)=[−12]. To find out what this vector means in our coordinate system, we must compute [v](e→1,e→2).

We can write [I](b→1,b→2)(e→1,e→2)[v](b→1,b→2)=[v](e→1,e→2).

To summarize, we can write the same equation in multiple ways:

Equation Description
−1b→1+2b→2=−1[21]+2[−11]=[−41] Linear combination of Jennifer's basis vectors
[b→1b→2][−12]=[2−111][−12]=[−41] Matrix multiplication
[I](b→1,b→2)(e→1,e→2)[v](b→1,b→2)=[v](e→1,e→2) Change of coordinate equation
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