Yes, f is a linear transformation.
To show that f is a linear transformation, we need to show that it satisfies the following two properties:
Closure under addition: For all vectors u and v in the domain of f, and for all scalars c and d, f(cu + dv) = cf(u) + df(v).
Homogeneity: For all scalars c and for all vectors u in the domain of f, f(cu) = cf(u).
Let's check each of these properties in turn.
Closure under addition:
f(cu + dv) = 8(cu + dv) - 2(cu + dv) = 6cu + 6dv = c(8u - 2v) + d(8v - 2u) = cf(u) + df(v).
Homogeneity:
f(cu) = 8(cu) - 2(cu) = 6cu = c(8u - 2v) = cf(u).
Since f satisfies both closure under addition and homogeneity, we can conclude that it is a linear transformation.
Question
Let f : R? → R be defined by f((x,y)) = 8y – 2x. Is f a linear transformation? a. f(x1, Yı) + (*2, Y2)) . (Enter x1 as x1, etc.) f(r1, Y1)) + f((x2, Y2) ) = + Does f((x1, Y1) + (x2; Y2)) = f((¤1, Y1)) + f((x2, Y2)) for all (21, Y1), (x2, Y2) E R?? choose b. f(c{x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f({x, y))) for all c E R and all (x, y) E R? choose c. Is f a linear transformation? choose