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

Haven 2020-12-15 Answered
Prove whether f:RR defined by f(x)=4x2 is a linear transformation.
Prove whether f:RR defined by f(x)=2x is a linear transformation.
Which one equivalent to the linear transformation T:RR defined by T(1)=2?
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Expert Answer

SchepperJ
Answered 2020-12-16 Author has 96 answers
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)=4x2 and let f(y)=4y2
Now we check if f(x+y)=f(x)+f(y)
f(x+y)=4(x+y)2=4x+4y2
And,
f(x)+f(y)=4x2+4y2=4x+4y4
f(x+y) is not equal to f(x)+f(y)
Hence f(x)=4x2 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:RR,f(x)=2x is equivalent to T:RR,T(1)=2.
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