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

Beginner2022-01-08Added 41 answers

That is let $\{T,U\}$ 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,

$0=(\alpha T+\beta U)\left(x\right)$

$=\alpha T\left(x\right)+\beta U\left(x\right)$

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, $\{T,U\}$ is linearly independent subset of L (V, W).
Hence proved.

Let Assume that a #0 then there exist xeV such that x e kemel of T.

Let

It is known that,

This can also be written as follows.

Which implies y is the image of U.

So,

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