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Answered

2022-10-16

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Under what conditions, if any, does a vector $(u,v,w)$ lie under the image of T?

I'm being asked to consider $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ given by the formula:

$(u,v,w)=T(x,y,z)=(x-y,y-z,z-x)$

Then, I'm asked under what conditions, if any, does a vector (u,v,w) lie in the image of T? lie under the image of T?

If my understanding is correct, the term "image" refers to the range of a transformation, meaning any element mapped to by T within the codomain is the image. lie under the image of T?

It looks to me that any value for u, v, or w will be acceptable under T, but I don't know how to validate that. I think I need some sort of equation that will prove that any input will produce a valid output. Is this a valid line of thinking, or am I on the wrong track? lie under the image of T?

My question is: How do I prove that any value is acceptable for a valid output?

Answer & Explanation

Amaya Vance

Expert

2022-10-17Added 6 answers

$(u,v,w)$ will lie in the image of T precisely if it is a linear combination of the columns of T′s matrix. Namely, $(u,v,w)\in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{(1,-1,0),(0,1,-1),(-1,0,1)\}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{(1,-1,0),(0,1,-1)\}$

That is, the vector must lie on the plane $x+y+z=0$

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