Linear Algebra Help: Is L:R^2 -> R^2 a Linear Transformation?

AI Thread Summary
The discussion centers on determining whether the function L: R^2 -> R^2, defined by L(x,y) = (x-1, y-x), qualifies as a linear transformation. To confirm this, two key properties of linear transformations must be verified: additivity and homogeneity. Participants are encouraged to test these properties using arbitrary elements from R^2 and a scalar. Resources are provided for further clarification on the definitions and properties of linear maps. Ultimately, the conclusion hinges on whether both properties hold true for the given function.
esler21
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Is L:R^2 - ->R^2 defined by L(x,y) = (x-1,y-x) a linear transformation? Explain why or why not.



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esler, I suggest you take this one over to the math section of the forum.
 
Apply the properties of a linear transformation:
http://en.wikipedia.org/wiki/Linear_map#Definition_and_first_consequences

There are two properties to verify, as outlined in the link. For the first one, take two arbitrary elements of R2, you can call them (x1, y1) and (x2, y2), and see if they satisfy the first property. You may use the usual point wise addition for the ordered pairs.
Then take an arbitrary scalar to see if the second property is satisfied.

You might also find this page useful:
http://ltcconline.net/greenl/courses/203/Vectors/linearTransRn.htm
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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