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(a) rotates each vector through the angle .
(b) is the orthogonal projection on the -plane.
(c) is multiplication by the matrix
Solution (a) , so T is one-to-one.
contains nonzero vectors, so T is not one-to-one.
From Example 5 of Section 8.2,
, so T is not one-to-one.
Solution (b)
From Example 4 of Section 8.2,
Solution (c)
From Example 7 of Section 8.2,
In the special case where T is a linear operator on a finite-dimensional vector space, a fourth equivalent statement can be
added to those in Theorem 8.3.1.
THEOREM 8.3.2
If V is a finite-dimensional vector space, and is a linear operator, then the following are equivalent.
(a) T is one-to-one.
(b) .
(c) .
(d) The range of T is V; that is, .
