1. Prove directly from the axioms for a vector space that a0^{→} = 0^{→} for any scalar a. Hint: you might want to analyze how we proved a similar fact inWeek4_Lect2_HW.pdf(the proof is different from the one in the Leon’s book).Remark. Of course, this fact is not hard to believe when we think of the concrete exs of vector spaces that we’ve seen. The point is: is it true in any vector space, including exs we haven’t seen or imagined yet?
Your proof must be logically sound. A common mistake is when one tries to prove a statement “A” by assuming “A”. See also http://en.wikipedia.org/wiki/Circular_reasoning .
2. Prove directly from the axioms for a vector space that if av^{→} = 0^{→} for some scalar a, then either a = 0^{→} orv^{→} = 0^{→}.
HINT: Sometimes, when you need to show that something could not happen, a good way to argue is to suppose that it could and then arrive to a logical contradiction.
3. Determine if the following are subspaces of R^{2}. Justify your answer. If a set is not a subspace, indicate what axiom is not satisfied.
a)
b)
4. Let R^{4X4} be the vector space of all 4X4 matrices. Show that the set of skew-symmetric
Matrices o(4) {A ≡ R^{4X4 }|A^{T }= -A}
is a subspace of R.
Do you really need to prepare a typical 4X4 matrix A, then it’s transpose, etc? Can you just use the properties of matrix arithmetics, transpose, etc.?
5. Determine if the following are subspaces of P_{3}. Justify your answer. If a set is not a subspace, indicate what axiom is not satisfied.
(a) The set of polynomials of degree 2{a_{2}x^{2} + a_{1}x + a_{0}|a_{i} ≡ R, a_{2} ≠ 0}:
(b) The set of polynomials p(x) such that p(1) = 0.
(c) The set of polynomials p(x) such that p(1) = 1.
6. Prove that if U and V are subspaces of a vector space W, then their sum defined as U + V ={^{→}w ≡ W|^{→}w =^{→}u +^{→}v where ^{→}u 2 U^{→}v≡ V } is also a subspace of W.
7. Let A = Determine the nullspace N(A).