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Problem Set 1
Math 423, Section 200, Spring 2015
Due: Friday, January 30.
Review Chapters 1 and 2 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Friday, January 30. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. For each of the following subsets of F3 , determine whether it is a subspace of F3 .
a. {(x1 , x2 , x3 ) ∈ F3 | x1 + 2x2 + 3x3 = 0}
b. {(x1 , x2 , x3 ) ∈ F3 | x1 + 2x2 + 3x3 = 4}
c. {(x1 , x2 , x3 ) ∈ F3 | x1 x2 x3 = 0}
d. {(x1 , x2 , x3 ) ∈ F3 | x1 = 5x3 }
2. Suppose U = {(x, x, y, y) ∈ F4 | x, y ∈ F}. Find a subspace W of F4 such that F4 = U ⊕ W.
3. Prove or give a counterexample: If U1 , U2 and W are subspaces of a vector space V, and
U1 + W = U2 + W then U1 = U2 .
4. Prove or give a counterexample: If v1 , v2 , . . . , vm are linearly independent vectors in V, then
5v1 − 4v2 , v2 , . . . , vm
is linearly independent.
5. Find a number t ∈ R such that
(3, 1, 4), (2, −3, 5), (5, 9, t)
is not linearly independent in R3 .
6. Prove that F∞ = {(x1 , x2 , x3 , . . . ) | xi ∈ F for i = 1, 2, 3, . . . } is infinite dimensional.
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7. Prove or give a counterexample: If v1 , v2 , v3 , v4 is a basis of V and U is a subspace of V such
that v1 , v2 ∈ U and v3 < U, v4 < U, then v1 , v2 is a basis of U.
8. Let U = {p ∈ P4 (F) | p(2) = p(6)}.
a. Find a basis of U.
b. Extend the basis of part a. to a basis of P4 (F).
c. Find a subspace W of P4 (F) such that C5 = U ⊕ W.
9. Suppose that U and W are both five-dimensional subspaces of R9 . Prove that U ∩ W , {0}.
10. Suppose U and W are both four-dimensional subspaces of C6 . Prove that U ∩ W contains two
vectors such that neither one is a scalar multiple of the other.
11. Suppose U1 , . . . , Um are finite dimensional subspaces of a vector space V such that U1 +· · ·+Um
is a direct sum. Prove that U1 ⊕ · · · ⊕ Um is finite dimensional and
dim(U1 ⊕ · · · ⊕ Um ) = dim(U1 ) + · + dim(Um ).
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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