Dr. Marques Sophie
Office 519
Algebra II
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #2
Due Monday october 7th in Class
Exercise 1(?): (4 points)
Decide whenever the following set of vector are linearly dependent or independent (Give
reasons)
1. 1, 1 + x, 1 + x + x2 in R2 [x]
2. (1, 1, 0, 0),(1, -1, 0 , 0), (1, 0, 0, 0) in R4 ;
3. (1, 2, 3), (2, 3, 4), (3, 4, 5) in R3
4. v, λv + w in some vector space V over some field F , for some arbitrary linearly
independent vectors v and w in V and λ ∈ F .
Exercise 2(??): (4 points)
Let H and G be vector spaces over a field F .
1. Prove that H ∩ G is a vector space over F .
2. Prove that H ∪ G is a vector space over F if and only if H ⊂ G or G ⊂ H.
3. Let H = span((2, 5, 3)) and G = span((2, 0, 5)), find H ∩ G in R3 .
Exercise 3(?): (3 points)
Is it possible to find a basis of R3 containing the vectors (1, 2, 5) and (0, 1, 2)? If yes,
find it. Find the coordinates of the vector (3, 2, 1) relative to this basis.
Exercise 4(??) (4,5 points):
Let t a homomorphism of vector space from V to W over some field F .
1. Using theorems of the class prove the rank-nullity theorem, dim(V ) = dim(ker(t))+
dim(Im(t)).
2. Let t a homomorphism from V = F 4 and W = R2 such that
Ker(t) = {(x1 , x2 , x3 , x4 ) ∈ F 4 : x1 = 5x2 and x3 = 7x4 }
Prove that t is surjective.
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3. Prove that there does not exist homomorphism t of vector space from F 5 to F 2
such that ker(t) = {(x1 , x2 , x3 , x4 , x5 ) ∈ F 5 : x1 = 3x2 and x3 = x4 = x5 }.
Exercise 5(??) (5 points):
Suppose that {vi : i = 1, ..., m} is an orthonormal set in Rn , i.e.,
1 if i = j
vi .vj =
0 if i 6= j
1. Show that {vi ; i = 1, ..., m} are linearly independent.
2. Show that, if there exists a nonzero vector v such that v.vi = 0 for i = 1, ..., m,
then we can enlarge {vi : i = 1, ..., m} to a bigger orthonormal set.
3. Challenge: Suppose n ≥ 2, let v1 ∈ Rn construct a non zero vector v2 such that
v1 .v2 = 0. (Hint: consider a element v linearly independent with v1 and construct
v2 with this element, use question 4. of Exercise 1.) More generally, if we take
v1 , ..., vm an orthonormal set m < n, construct a non zero vector vm+1 such that
vi .vm+1 = 0 for any i.
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