Seettiffrourfk6

2022-10-16

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Under what conditions, if any, does a vector $\left(u,v,w\right)$ lie under the image of T?
I'm being asked to consider $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ given by the formula:
$\left(u,v,w\right)=T\left(x,y,z\right)=\left(x-y,y-z,z-x\right)$
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?

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Amaya Vance

Expert

$\left(u,v,w\right)$ will lie in the image of T precisely if it is a linear combination of the columns of T′s matrix. Namely, $\left(u,v,w\right)\in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{\left(1,-1,0\right),\left(0,1,-1\right),\left(-1,0,1\right)\right\}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{\left(1,-1,0\right),\left(0,1,-1\right)\right\}$
That is, the vector must lie on the plane $x+y+z=0$

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