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Professor Dave here, lets change things up. Now that we know about basis vectors, we are ready to learn how to change from one basis to another, which can be useful in certain situations. This can help express a problem in a way that is more easily solved, so lets learn how to do this now. We saw in the previous tutorial that having vectors of length one in each of the x, y, and z coordinates, is in fact a basis. For this example consider the vector space R2, which contains vectors in the two-dimensional x-y space. Our most simple basis here will be a unit vector in the x direction, i, which can be expressed as (1, 0), and a unit vector in the y direction, j, which can be expressed as (0, 1). Because we know this is a basis, we can write any vector in R2, represented by v, as a linear combination of i and j. Lets say that v equals (v1, v2). We can easily write this as v1 times (1, 0) plus v2 times (0, 1). Or equivalently, v equals v1(i) + v2(j). We have written our vector in terms