Page 208 - A Course in Linear Algebra with Applications
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192 Chapter S ix: Linear Transformations
12. (The third isomorphism theorem). Let U and W be sub-
spaces of a vector space V such that W C. U. Prove that U/W
is a suhspace oiV/W and that (V/W)/(U/W) ~ V/U. [Hint:
define a function T : V/W -> V/U by the rule T(v + W) =
v + ?7. Show that T is a well defined linear transformation and
apply 6.3.7].
13. Explain how to define a power T m of a linear operator T
on a vector space V, where m > 0. Then show that powers of
T commute.
