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Exercise 2.1.7

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For property 1, we have $T (0) = T (0 x) = 0 T (x) = 0$ ,where $x$ is an arbitrary element in $V$ . For property 2, if $T$ is linear, then $T (cx + y) = T (cx) + T (y) = cT (x) + T (y)$ ; if $T (cx + y) = cT (x) + T (y)$ , then we may take $c = 1$ or $y = 0$ and conclude that $T$ is linear. For property 3, just take $c = - 1$ in property 3. For property 4, if $T$ is linear, then

T (\sum_{i = 1}^{n} a_{i} x_{i}) = T (a_{1} x_{1}) + T (\sum_{i = 1}^{n - 1} a_{i} x_{i}) = \dots = \sum_{i = 1}^{n} T (a_{i} x_{i}) = \sum_{i = 1}^{n} a_{i} T (x_{i});

if the equation holds, just take $n = 2$ and $a_{1} = 1$ .

jephianlin

2011-06-27 00:00

Exercise 2.1.7

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