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Transcript
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 .
1
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
2