If we have basis {e1=(2,-1,-1),e2=(3,1,1),e3=(-2,-1,-2)} and basis {u1=(-3,1,2),u2=(1,1,3),u3=(-2,-2,-1)} The question is prove that e1,e2,e3 and u1,u2,u3 forms basis of R³ And find the transition matrix from first basis to the second one. And if we have the coordinates of vectors [X]e=(-2,2,-2) and [Y]u=(2,-1,1) in one basis, find the coordinates of these vectors in the another basis. I proved correctly and I found the transition matrix, but how can i find the coordinates of vectors [X]e and [Y]u in another basis? I think that I should use the transition matrix. Please can any one give a hint Hint the transition matrix I got: Transition matrix C=B.A^-1, where B is the matrix of vectors u1,u2,u3 as a columns of thier coordinates, and A is the matrix of coordinates of vectors e1,e2,e3
(-2/5 8/5 3/5)
(2/5 -8/5 7/5)
(1 1 -1)
You are on the right track. If $M_{E\to U}$ is the transition matrix from $E$ to $U$, and you have the coordinates of $X$ in the base $E$, to get them on the base $U$ you just need to multiply $$[X]_U=M_{E\to U}\cdot[X]_E.$$ To do the reverse change of basis, you just need the transition matrix from $U$ to $E$, but that's nothing but the inverse of the previous matrix: $$M_{U\to E}=M_{E\to U}^{-1}.$$
Just to clarify, I am guessing that you are calculating $M_{E\to U}$ as a matrix whose columns are the coordinates of $e_i$ in the $U$ basis. Note that the coordinate vectors $[X]_E$ and $[Y]_U$ must be written in column form so you be able to make the product.
You could also have calculated $M_{E\to U}$ as a matrix whose rows are the coordinates of $e_i$ in the $U$ basis. Then, the vectors in the product would be written in row form and the product would be: $$[X]_U=[X]_E\cdot M_{E\to U}$$