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Exercise 2.3.8

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In general we may set $T_{1}, T_{2} \in L (X, Y)$ and $U_{1}, U_{2} \in L (W, X)$ , and $S \in (V, W)$ , and thus we have the following statements.

1.: $T_{1} (U_{1} + U_{2}) = T U_{1} + T U_{2}$ and $(U_{1} + U_{2}) T = U_{1} T + U_{2} T$ .
2.: $T_{1} (U_{1} S) = (T_{1} U_{1}) S$ .
3.: $T I_{X} = I_{Y} T = T$ .
4.: $a (T_{1} U_{1}) = (a T_{1}) U_{1} = T_{1} (a U_{1})$ for all scalars $a$ .

To prove this, just map arbitrary vectors in the domain by linear transformations and check whether the vectors produced by different transformations meet.

jephianlin

2011-06-27 00:00

Exercise 2.3.8

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