Download Math102 Lab8

Math102 Lab8
1. Determine whether the following sets are vector spaces under the given operations. For those that are not vector spaces list all axioms that fail to hold.
a 1
(a) The set of all 2 × 2 matrices of the form
with the standard matrix
1 b
addition and scalar muptiplication.
(b) The set of all pairs of real numbers of the form (1, x) with the operations
(1, y) + (1, y 0 ) = (1, y + y 0 ) and k(1, y) = (1, ky).
2. Use theorem 5.2.1 to determine which of the following are subspaces of M22
(a) all 2 × 2 matrices with integer entries
a b
(b) all matrices
where a + b + c + d = 0
c d
3. Express (−9, −7, −15) as linear combinations of u = (2, 1, 4), v = (1, −1, 3) and
w = (3, 2, 5)
4. Is the set {(3, 8, 7, −3), (1, 5, 3, −1), (2, −1, 2, 6), (1, 4, 0, 3)} of vectors in R4 linearly independent?
5. Show that the vectors v1 = (0, 3, 1, −1), v2 = (6, 0, 5, 1) and v3 = (4, −7, 1, 3)
form a linearly dependent set in R4 .
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Solution
1. Look at the list of axioms on page 222.
(a) Axioms 1,4,5 and 6 fail.
(b) This set is a vector space with the zero element (1, 0) and the identity
element as (1, 1).
2. Recall Theorem 5.2.1: If W is a set of one or more vectors from a vector space
V , then W is a subspace of V if and only if the following conditions holds:
(i) If u and v are vectors in W , then u + v is in W .
(ii) If k is any scalar and u is any vector in W , then ku is in W .
(a) If you take k as any real number then (ii) of the theorem fails, hence this
set is not a subspace of M22
(b) Both (i) and (ii) holds for this set, hence it is a subspace of M22
3. (−9, −7, −15) = −2u + v − 2w
4. Yes it is linearly independent.
5. One can show that v3 =
dependent set in R4 .
2
v
3 2
− 37 v1 , hence this shows they form a linearly
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