Question

Let T : R^3 -> R^2, T(x, y, z) = (2x + y + z, y - 3z)
Show that T is a linear transformation.

Answer 1

let u=(x,y,z) and u'=(x',y',z') be any element of R^3, and c be any constant.
T(u+u')=T(x+x',y+y',z+z')=(2(x+x')+(y+y')+(z+z'),(y+y')-3(z+z'))=(2x+y+z+2x+y+z,y-3z+y'-3z')=(2x+y+z,y-3z)+(2x'+y'+z',y'-3z')=T(u)+T(u')
T(cu)=(2xc+cy+cz,cy-3cz)=c(2x+y+z,y-3z)=cT(u)
since T(u+u')=T(u)+T(u') and T(cu)=cT(u), then it is a linear transformation

Answer 2

and the induced matrix is:
\[\left[\begin{matrix} 2 & 0 \\ 1 & 1 \\ 1& -3\end{matrix}\right]\]

Answer 3

Find a matrix C for T with respect to the standard bases for R^3 and R^2

Answer 4

so let A denote the induced matrix
T(x,y,z)=A[x,y,z]^T
where [x,y,z]^T is the transpose of [x,y,z]
sorry the induced matrix should be A^T in that case, or the transpose of A