Let V {\displaystyle {}V} and W {\displaystyle {}W} be finite-dimensional K {\displaystyle {}K} -vector spaces. Let v = v 1 , … , v n {\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}} and u = u 1 , … , u n {\displaystyle {}{\mathfrak {u}}=u_{1},\ldots ,u_{n}} be bases of V {\displaystyle {}V} , and w = w 1 , … , w m {\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{m}} and z = z 1 , … , z m {\displaystyle {}{\mathfrak {z}}=z_{1},\ldots ,z_{m}} bases of W {\displaystyle {}W} . Let M u v {\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}} and M z w {\displaystyle {}M_{\mathfrak {z}}^{\mathfrak {w}}} be the transformation matrices. What is the transformation matrix for the base change from the basis ( v 1 , 0 ) , … , ( v n , 0 ) , ( 0 , w 1 ) , … , ( 0 , w m ) {\displaystyle {}(v_{1},0),\ldots ,(v_{n},0),(0,w_{1}),\ldots ,(0,w_{m})} to the basis ( u 1 , 0 ) , … , ( u n , 0 ) , ( 0 , z 1 ) , … , ( 0 , z m ) {\displaystyle {}(u_{1},0),\ldots ,(u_{n},0),(0,z_{1}),\ldots ,(0,z_{m})} of the product space V × W {\displaystyle {}V\times W} ?
and 
be
finite-dimensional
-vector spaces.
Let
and 
be
bases
of , and
and 
bases of . Let
and 
be the
transformation matrices.
What is the transformation matrix for the base change from the basis 
to the basis 
of the
product space
?
