Exercise 2.2.13

Question

If $T$ and $U$ are nonzero linear transformations from $V$ to $W$, then they form a linearly independent set provided $Range(T)\cap Range(U)=\{0\}$.

Jephian C.-H. Lin · Accepted Answer

Suppose, by contradiction, that $cT=U$ for some $c$. Since $T$ is not zero mapping, there is some $x\in V$ and some nonzero vector $y\in W$ such that $T(x)=y
eq 0$. But thus we have $$y=\frac{1}{c}cy=\frac{1}{c}U(x)=U(\frac{1}{c}x)\in R(U).$$ This means $y\in R(T)\cap R(U)$, a contradiction.

Exercise 2.2.13

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Exercise 2.2.13

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