Let T R3 rightarow R2 be the linear transformation that has

Let T: R^3 rightarow R^2 be the linear transformation that has the matrix [2 3 1 1 2 1] relative to the bases alpha = {(1,-1,1),(0,1,0),(1,0,0)} of R^3 and beta = {(3,2),(2,1)} of R^2. Find a formula for T((x,y,z)) for any (x,y,z) R^3.

Solution

In R³ (x,y,z) be a vector and corresponding to the given basis {(1,-1,1) , (0,1,0),(1,0,0)} we can represent (x,y,z) as ,

(x,y,z) = z*(1,-1,1) + (y+z)*(0,1,0) + (x-z)*(1,0,0)

So the co-ordinate corresponding to the given basis is {z,(y+z),(x-z)}

Therefore,

T((x,y,z) ) = T(z*(1,-1,1) + (y+z)*(0,1,0) + (x-z)*(1,0,0) ) = z*T((1,-1,1)) + (y+z)*T((0,1,0)) + (x-z)*T((1,0,0)) = z*(2,1) + (y+z)*(3,2) + (x-z)*(1,1) = ( x+3y+4z , x+2y+2z ).............This is the formula.