Seettiffrourfk6

Seettiffrourfk6

Answered

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 ( u , v , w ) lie under the image of T?
I'm being asked to consider T : R 3 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?

Do you have a similar question?

Recalculate according to your conditions!

Answer & Explanation

Amaya Vance

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 ) s p a n { ( 1 , 1 , 0 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 1 ) } = s p a n { ( 1 , 1 , 0 ) , ( 0 , 1 , 1 ) }
That is, the vector must lie on the plane x + y + z = 0

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