Final answer:
Yes, f(x) = <x, 6x², 7x> is a linear transformation.
Step-by-step explanation:
For a function to be a linear transformation, it must satisfy certain properties. One of these properties is that it preserves addition. Let's check if f(x) = <x, 6x², 7x> preserves addition.
Let's say we have two vectors u = <a, b, c> and v = <d, e, f> in &R³. The sum of these vectors is u + v = <a+d, b+e, c+f>.
Now let's calculate f(u) and f(v).
f(u) = f(a, b, c) = <a, 6a², 7a>
f(v) = f(d, e, f) = <d, 6d², 7d>
We can see that f(u) + f(v) = <a+d, 6a²+6d², 7a+7d> which is equal to f(u+v) if and only if a+d = a+d, 6a²+6d² = 6(a²+d²), and 7a+7d = 7(a+d).
Therefore, f(x) = <x, 6x², 7x> is a linear transformation as it preserves addition.