Let V and W be vector spaces, and let T



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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) 

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Beginner2022-01-08Added 41 answers

That is let {T,U} is not linearly independent subset of L(V,W) then there exist some α,β not both such that αT. βU=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(x)0.
It is known that,
This can also be written as follows.
Which implies y is the image of U.
So, yImTImU{0} which proves the contrapositive. Thus, {T,U} is linearly independent subset of L (V, W). Hence proved.

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