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Unformatted text preview: 4) If A and B are n × n matrices over the ﬁeld F show that tr(AB ) = tr(BA). Then show that similar matrices
have the same trace.
4) If A and B are complex matrices, show that AB − BA = I is impossible. 2 −2
. Let W be the subspace of V consisting of all A such that AB = 0.
Let f be a linear functional on V which is in the annihilator of W and such that f (I ) = 0 and f (C ) = 3 where
. Find f (B ).
4) Let V = R2×2 and let B = 4) Let V be a ﬁnite dimensional vectorspace over a ﬁeld F and let W be a subspace of V . If f is a linear functional
on W prove that there exists a linear functional g on V such that g (x) = f (x) for all x ∈ W .
4) Let F be a ﬁeld of characteristic 0 and let V be a ﬁnite dimensional vectorspace over F . If a1 , · · · , am are
ﬁnitely many vectors in V , each not the zero vector, prove that there is a linear functional f on V such that
f (ai ) = 0, for i = 1, · · · , m.
4) Let V be the vectorspace of n × n matrices over a ﬁeld F and let W0 be the subspace of V spanned by matrices
of the form AB − BA. Show that W0 is exactly the space of matrices with trace 0.
4 Let S be a set, F a ﬁeld, and V (S, F ) the space of all functions from S into F. Let W be any n-dimensional
subspace of V (S, F ). Show that there exist n points x1 , · · · , xn in S and n functions f1 , · · · , fn in W such that
fi (xj ) = δij .
4 Let F be a ﬁeld a let f be the linear functional on F 2 deﬁned by f (x, y ) = ax + by . Let g = T t f . Find g for
(a) T (x, y ) = (y, 0).
(b) T (x, y ) = (−y, x).
(c)T (x, y ) = (y, 0).
4) Let V be a ﬁnite dimensional vector space over the ﬁeld F and let T be a linear operator on V . Let c be a scalar
and suppose there is a non-zero vector a ∈ V such that T a = ca. Prove that there is a non-zero linear functional f
on V such that T t f = cf .
4) Let A be an n × m matrix over R. Show that A = 0 iﬀ tr AAT = 0. 1 ...
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This document was uploaded on 01/25/2012.
- Spring '09