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Vector Space Theorems and Matrices

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2. Use Theorem 5.2.1 to determine which of the following are subspaces of M22.

Thm 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 hold.
(a) If u and v are vectors in W, then u + v is in W.
(b) If k is any scalar and u is any vector in W, then ku is in W.

a) all 2 x 2 matrices with integer entries
b) all matrices a b where a + b + c + d = 0
c d

c) all 2 x 2 matrices A such that det(A) = 0
d) all matrices of the form a b
0 c

e) all matrices of the form a a
-a -a

18. Prove Theorem 5.2.4.

Thm 5.2.4: If S = {v1, v2, ..., vr}and S' = {w1, w2, ..., wk} are two sets of vectors in a vector space V, then

span{v1, v2, ..., vr}= span{w1, w2, ..., wk}

if and only if each vector in S is a linear combination of those in S' and each vector in S' is a linear combination of those in S.

22. Show that the set of continuous functions f = f(x) on [a, b] such that

The intergral from a to b f(x)dx = 0

is a subspace of C[a, b].

See attached file for full problem description.