Suppose you have a base {e_1, e_2, e_3} in R_3. You don't know anything else about them. Is it possible, if you are given another base {u_1, .. , u_3} to give a change-of-base matrix from the e_i's to the u_i's???

 

Thanks!

yes. write u_i in terms of the e_i, presumably one is given in terms of the other, even in a general form.

That's how I did it in the Linear Algebra course.

 

Basically:

u_1 = a_{11} e_1 + a_{21} e_2 + a_{31} e_3

u_2 = a_{12} e_1 + a_{22} e_2 + a_{32} e_3

u_3 = a_{13} e_1 + a_{23} e_2 + a_{33} e_3

 

and then your change of basis matrix is (a_{ij}).

 

I think. It's been a while since I've done this

Up to transpose that is correct. Though I can never remember which way round it is. The OP should do the mental maths to figure out which way round it is.