zakinutuzi

2022-01-07

Let T and U be nonzero linear transformations from V into W, and let V and W be vector spaces. Show that T, U is a linearly independent subset of L if R(T)

zesponderyd

That is let $\left\{T,U\right\}$ is not linearly independent subset of L(V,W) then there exist some $\alpha ,\beta$ not both such that $\alpha T$. $\beta U=0$, where 0 is the transformation that takes every element of V to zero.
Let Assume that a #0 then there exist xeV such that x e kemel of T.
Let $y=T\left(x\right)\ne 0$.
It is known that,
$0=\left(\alpha T+\beta U\right)\left(x\right)$
$=\alpha T\left(x\right)+\beta U\left(x\right)$
This can also be written as follows.
$y=T\left(x\right)$
$=\frac{-\beta }{\alpha }U\left(x\right)$
$=U\left(\frac{-\beta }{\alpha }\left(x\right)\right)$
Which implies y is the image of U.
So, $y\in ImT\cap ImU\ne \left\{0\right\}$ which proves the contrapositive. Thus, $\left\{T,U\right\}$ is linearly independent subset of L (V, W). Hence proved.

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