Consider two bases for R2

Question

 Consider two bases for R2

 

B= { [ 1 ] , [-1] }  and B' ={ [ 2 ], [ 4 ] }

          3       -1                       2      -1

 

1) Find the transition matrix from B to B' .

2) Vector u has coordinates (−5,−19) in the basis B. Find its coordinates in the basis B' .

    

 

Answer 1

The transition matrix from B to B' will be a matrix T such that B' = T x B.  (Of course here x is matrix multiplication).  So to find your transition matrix, you need to to know B^{-1}, that means B inverse, since B x B^{-1} = T x B x B^{-1} = T.

 

Once you find T, the coordinates of u in B' will be T x u. 

 

I guess you know how to multiply matrices, right?