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Unformatted text preview: Quiz #1
1. Label the following statements as being true or false. No veriﬁcation
is needed.
(a) A vector space may have more than one zero vector.
(Solution) False.
(b) If S is a subset of a vector space V , then span(S) equals the
intersection of all subspaces of V that contain S.
(Solution) True.
(c) Subsets of linearly dependent sets are linearly dependent.
(Solution) False.
(d) The dimension of Mm×n (F ) is m + n.
(Solution) False.
(e) Let T : V → W be a linear transformation of vector spaces. Then
T is onetoone if and only if N (T ) = {0}.
(Solution) True.
2. Determine whether the following set is a basis for R3 or not.
{(1, −3, −2), (−3, 1, 3), (−2, −10, −2)}.
(Solution) The above set is not a basis for R3 since they are linearly
dependent; 1 −3 −2
det −3 1 −10 = 0.
−2 3
−2
3. A matrix A is called skewsymmetric if At = −A. The set of all skewsymmetric 3 × 3 matrices is a subspace W of M3×3 (R) (You need NOT
prove it). What is the dimension of W ?
(Solution) Let a b c
A = d e f .
g h i 1 be a skewsymmetric matrix. Then, since At = −A,
d = −b, g = −c, h = −f. a = e = i = 0,
Therefore, 0 1 0
0 0 1
0 0 0
A = b −1 0 0 + c 0 0 0 + f 0 0 1 .
0 0 0
−1 0 0
0 −1 0
Thus, the subset 0 0 1
0 0 0 0 1 0
−1 0 0 , 0 0 0 , 0 0 1 0 0 0
−1 0 0
0 −1 0
is a basis for W , and hence dim W = 3.
[A generalized version] A matrix A is called skewsymmetric if
At = −A. The set of all skewsymmetric n × n matrices is a subspace
W of Mn×n (R). What is the dimension of W ?
(Solution) Let A be skewsymmetric. Then
(At )ij = (−A)ij
for all i, j = 1, . . . , n. So
Aji = −Aij .
Therefore Aii = 0 for all i = 0, . . . , n and
n A= Aij (Eij − Eji ). Aij Eij =
i,j=1 1≤j<i≤n Thus, the set
{Eij − Eji  1 ≤ j < i ≤ n}
is a basis for W and
n
2 dim W = 2 = n(n − 1)
.
2 ...
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 Spring '06
 Holtz
 Linear Algebra, Algebra, Vector Space, aij Eij, linearly dependent sets

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