Question

(1 point) Let f:R→R3 be defined by f(x)=⟨6x,−6x,−9x⟩. Is f a linear transformation? f(x+y)= 6(x+y),-6(x+y),-9(x+y) . f(x)+f(y)= (6x,-6x,-9x) + (6y,-6y,-9y) . Does f(x+y)=f(x)+f(y) for all x,y∈R? f(cx)= (6(cx),-6(cx),-9(cx)) . c(f(x))= c ( 6cx,-6cx,-9cx ). Does f(cx)=c(f(x)) for all c,x∈R? Is f a linear transformation?

Answer 1

Answer:

f(x+y) = f(x)+f(y)

f(cx)=c(f(x)), for ∀x,c ∈ R

Therefore f is a linear transformation.

Step-by-step explanation:

Linear transformation:

Let a and b be two vector space over some field R. A function f: A→B  is said to be linear transformation if for any two vectors a, b∈ A and any scalar c∈ R such that

(i)f(a+b) = f(a)+ f(b)

(ii) f(ca)= cf(a)

Here the function is f(x)= <6x,-6x,-9x>

Then f(y) = <6y,-6y,-9y>  [ putting x= y in the function]

(i)

f(x+y)

= <6(x+y),-6(x+y),-9(x+y)>

f(x)+f(y)

=<6x,-6x,-9x> + <6y,-6y,-9y>

=<6x+6y,-6x-6y,-9x-9x>

=<6(x+y),-6(x+y),-9(x+y)>

Therefore f(x+y) = f(x)+f(y)

(ii)

c(f(x))

=c<6x,-6x,-9x>

f(cx)

=<6cx,-6cx,-9cx>

=c<6x,-6x,-9x>

Therefore f(cx)=c(f(x)), for ∀x,c ∈ R

Therefore f is a linear transformation. Since it satisfies the two condition.