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has , whereas if but b is nonzero, then every point of the form
has , and if but a is nonzero, then every point of the form
has . Finally, in the degenerate case and , we have for every v in .
In each case, T fails to be one-to-one, so there can be no transformation from the plane to the real line that is both linear and
one-to-one.
Of course, even if , a linear transformation from V to W might not be one-to-one, as the zero
transformation shows.
Exercise Set 8.3
Click here for Just Ask!
In each part, find , and determine whether the linear transformation T is one-to-one.
1.
(a) , where
(b) , where
(c) , where
(d) , where
(e) , where
(f) , where
In each part, let be multiplication by A. Determine whether T has an inverse; if so, find
2.
(a)
