Change of Basis

Question

Define u_1=?(1,1,0)?^T,u_2=?(0,1,1)?^T,u_3=?(1,0,1)?^T and v_1=?(1,1,2)?^T,v_2=?(2,1,1)?^T,v_3=?(1,2,1)?^T, then obtain the transition matrix from (v1,v2,v3) to (u1,u2,u3) and also from (u1,u2,u3 ) to (v1,v2,v3). Also, find out the coordinates of the vector ?(1,1,1)?^T with respect to both the bases.

Please if you can show all the steps and explain. Thank you

Answer 1

Jorge,

 

Let me restate the information you provided.

 

The basis u = {u₁, u₂, u₃} is given with the vectors u₁ = (1,1,0), u₂ = (0,1,1), u₃ = (1,0,1). You also are given a second basis v = {v₁, v₂, v₃} with vectors v₁ = (1,1,2), v₂ = (2,1,1,), v₃ = (1,2,1).

 

The transition matrix from u to v is used to find coordinates of a vector in R³ w.r.t. the vectors in v, given the coordinates of the same vector w.r.t. u. The best way to do this is to use Gauss-Jordan elimination to find the matrix T. You can also do this without row operations, but you still need to solve three systems of linear equations, which amounts to the same thing.

 

Form the 3 x 6 matrix [v : u] obtained by writing the columns of v first (in proper order), followed by the columns of u. Once you form this matrix, you can use row reduction techniques to find the augmented matrix [I : P^-1]. The matrix P^-1 formed by the last three columns of this reduced matrix will be the transition matrix T from u to v. Its inverse is the transition matrix from v to u.

 

Please pay special attention to the order in which you list the columns of this matrix, otherwise you will get nonsense.

 

I will let you try this on your own. I could show you the answer, but I do not know how you are allowed to do row operations, whether by hand or with a computing device.

 

If you get stuck, please let me know and I will nudge you in the proper direction.