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January 21st, 2017, 06:56 PM   #1
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How to solve this problem?

Let L:R-->R be defined by L(u)=au+b where a and b are any real numbers. Find all values of a and b so that L is a linear transformation.

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Last edited by skipjack; January 21st, 2017 at 08:57 PM.
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January 21st, 2017, 07:04 PM   #2
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a linear transform $f(x)$ has the properties that

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

$f(\alpha x) = \alpha f(x),~\alpha \in \mathbb{R}$

apply these to $L(u)=a u + b$ to determine which $a,b$ make $L(u)$ linear
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January 21st, 2017, 07:36 PM   #3
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Quote:
Originally Posted by romsek View Post
a linear transform $f(x)$ has the properties that

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

$f(\alpha x) = \alpha f(x),~\alpha \in \mathbb{R}$

apply these to $L(u)=a u + b$ to determine which $a,b$ make $L(u)$ linear
$L(u+v) = a(u+v) + b = a u + a v + b$

$L(u)+L(v) = a u + a v + 2b$

so what must the value of $b$ be in order for $L$ to be linear?

when $b$ is set to this value the second property is trivial to prove.
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