Prove whether f:R→R defined by f(x)=4x−2 is a linear transformation. Prove whether f:R→R defined by f(x)=2x is a linear transformation. Which one equivalent to the linear transformation T:R→R defined by T(1)=2?

Question

Prove whether f:R→R defined by f(x)=4x−2 is a linear transformation.  Prove whether f:R→R defined by f(x)=2x is a linear transformation.  Which one equivalent to the linear transformation T:R→R defined by T(1)=2?

SchepperJ · Accepted Answer

Step 1  To prove that f(x) is linear transformation we have to prove that  f(x+y)=f(x)+f(y)  And af (x)=f(ax), a belongs to R  Here a is a scalar and R is the set of real numbers.  Step 2  f(x)=4x−2 and let f(y)=4y−2  Now we check if f(x+y)=f(x)+f(y)  f(x+y)=4(x+y)−2=4x+4y−2  And,  f(x)+f(y)=4x−2+4y−2=4x+4y−4  ⇒f(x+y) is not equal to f(x)+f(y)  Hence f(x)=4x−2 is not a linear transformation.  Step 3  f(x)=2x and f(y)=2y  f(x+y)=2(x+y)=2x+2y  And,  f(x)+f(y)=2x+2y  We get, f(x+y)=f(x)+f(y)  And,  af (x)=a(2x)=2ax=f(ax)  Both the properties are satisied, hence f(x)=2x is a linear transformation.  Step 4  f(x)=2x at x=1,f(1)=2  Hence f:R→R,f(x)=2x is equivalent to T:R→R,T(1)=2.

Prove whether {f}:mathbb{R}tomathbb{R} text{defined by} f{{left({x}right)}}={4}{x}-{2} is a linear transformation. Prove whether {f}:mathbb{R}tomathbb

Answered question

Answer & Explanation

New Questions in Integral Calculus