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(ii) If U and U are subspaces of a ﬁnite dimensional vector space V ,
prove that
dim(U ) + dim(U ) = dim(U ∩ U ) + dim(U + U ).
Solution. Take a basis of U ∩ U and extend it to bases of U and
of U .
(iii) If V is ﬁnite dimensional, prove that every subspace U of V is a
direct summand.
Solution. Extend a basis B of U to a basis B ∪ C of V , and show
that V = U ⊕ C .
4.20 (i) Prove that U ⊕ W is a vector space.
Solution. Absent. (ii) If U and W are ﬁnite dimensional vector spaces over a ﬁeld k ,
prove that
dim(U ⊕ W ) = dim(U ) + dim(W ).
Solution. If B is a basis of U and C is a basis of W , then B ∪ C
is a basis of U ⊕ V .
4.21 Assume that V is an n dimensional vector space over a ﬁeld k and that V
has a nondegenerate inner product. If W is an r dimensional subspace of
V , prove that V = W ⊕ W ⊥ (see Example 4.5). Conclude that dim(W ⊥ ) =
n − r.
4.22 Here is a theorem of Pappus holding in k 2 , where k is a ﬁeld. Let and m be
distinct lines, let A1 , A2 , A3 be distinct points on , and let B1 , B2 , B3 be
distinct points on m . Deﬁne C1 to be A2 B3 ∩ A3 B2 , C2 to be A1 B3 ∩ A3 B1 ,
and C3 to be A1 B2 ∩ A2 B1 . Then C1 , C2 , C3 are collinear.
State the dual of the theorem of Pappus.
Solution. Let P and Q be distinct points, let 1 , 2 , 3 be distinct lines
passing through P , and let m 1 , m 2 , m 3 be distinct liness passing through
Q . Deﬁne n 1 to be the line determined by 2 ∩ m 3 and 3 ∩ m 2 , n 2 to be the
line determined by 1 ∩ m 3 and 3 ∩ m 1 , and n 3 to be the line determined
by 1 ∩ m 2 and 2 ∩ m 1 . Then n 1 , n 2 , n 3 intersect in a point.
4.23 True or false with reasons.
(i) There is a solution to an n × n inhomogeneous system Ax = b if
A is a triangular matrix.
Solution. False.
(ii) Gaussian equivalent matrices have the same row space.
Solution. True. ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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