Question

A mapping T : R 3 −→ R 3 defined by T(x, y, z) = (x + y − z, 2xy, x + z + 1),(x, y, z) ∈ R 3 . Determine whether T is a linear transformation.​

Answer 1

It is not linear . Because, If u = (x, y, z) and v = (x’, y’, z’) be any two elements of R³ , then we can easily see that ;

(I) T(u + v) ≉ T(u) + T(v) . As (x+x’)(z+z’)]

=[(x+x’+y+y’), (x+x’-y-y’),(xz + xz’+x’z+x’z’)]

≉ (x+y, x-y, xz) + (x’+y’, x’-y’, x’z’)

= T(x, y, z) + T(x’, y’, z’) = T(u) + T(v) . Therefore, T is not linear. Now no need to show the homogeneity i.e.T(α.u) = α. T(u) etc.