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(ii) Every matrix over R is similar to inﬁnitely many different matrices.
Solution. False.
(iii) If S and T are linear transformations on the plane R2 that agree on
two nonzero points, then S = T .
Solution. True.
(iv) If A and B are n × n nonsingular matrices, then A + B is nonsingular.
Solution. False.
(v) If A and B are n × n nonsingular matrices, then AB is nonsingular.
Solution. True.
(vi) If k is a ﬁeld, then
{ A ∈ Matn (k ) : AB = B A for all B ∈ Matn (k )}
is a 1dimensional subspace of Matn (k ).
Solution. True.
(vii) The vector space of all 3 × 3 symmetric matrices over R is isomorphic to the vector space consisting of 0 and all f (x ) ∈ R [x ]
with deg( f ) ≤ 5.
Solution. True.
(viii) If X and Y are bases of a ﬁnite dimensional vector space over a
ﬁeld k , then Y [1V ] X is the identity matrix.
Solution. False.
(ix) Transposition Matm ×n (C) → Matn ×m (C), given by A → A T , is
a nonsingular linear transformation.
Solution. True.
(x) If V is the vector space of all continuous f : [0, 1] → R, then
1
integration f → 0 f (x ) d x is a linear transformation V → R.
Solution. True.
4.33 Let k be a ﬁeld, let V = k [x ], the polynomial ring viewed as a vector space
over k , and let Vn = 1, x , x 2 , . . . , x n . By Exercise 4.8, we know that
X n = 1, x , x 2 , . . . , x n is a basis of Vn .
(i) Prove that differentiation T : V3 → V3 , deﬁned by T ( f (x )) =
f (x ), is a linear transformation, and ﬁnd the matrix A = X 3 [T ]
X3
of differentiation.
Solution. Absent. ...
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 Fall '11
 KeithCornell

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