Download Homework #2

Last updated 9/17/2014
Math 355 Homework Problems
Math 355 Homework Problems #2
1. Let
S = {w1 , w2 , . . . , wk },
S + = {w1 , w2 , . . . , wk , v}
be two sets of vectors from a vector space V . Show that Span(S) = Span(S + ) if and only if v ∈ Span(S).
2. For a set B and linear operator A denote
A(B) = {A(b) : b ∈ B}.
Let V W with isomorphism L : V 7→ W . Show that:
(a) if the set {v1 , . . . , vk } ⊂ V is linearly independent, then so is the set {L(v1 ), . . . , L(vk )} ⊂ W .
(b) if S ⊂ V is a subspace with dim[S] = k, then there is a unique subspace T ⊂ W with dim[T ] = k
such that L(S) = T
(c) if T ⊂ W is a subspace with dim[T ] = `, then there is a unique subspace S ⊂ V with dim[S] = `
such that L−1 (T ) = S.
The importance of parts (b) and (c) is that they demonstrate that if V W , then the respective subspaces
are also isomorphic to each other.
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3. Let L1 : V1 7→ V2 and L2 : V2 7→ V3 be invertible linear maps. Show that (L2 L1 )−1 = L−1
1 L2 .
4. Let L : V 7→ V be a linear operator, and suppose that V is finite-dimensional. Denote
Lk = L ◦ L ◦ · · · ◦ L .
| {z }
k times
Note that Lk : V 7→ V for each k ≥ 1. Show that
(a) ker(Lj ) ⊂ ker(Lj+1 )
(b) if for some k, ker(Lj ) = ker(Lj+1 ), then ker(Ln ) = ker(Lk ) for all n ≥ k.
5. Let L : V 7→ V be a linear operator, and suppose that V is finite-dimensional. Show that
(a) Ran(Lj+1 ) ⊂ Ran(Lj )
(b) if for some k, Ran(Lk+1 ) = Ran(Lj ), then Ran(Ln ) = Ran(Lk ) for all n ≥ k.
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Math 355 Homework Problems
6. The first four Legendre polynomials are given by
f1 (x) = 1, f2 (x) = x, f3 (x) = 3x2 − 1, f4 (x) = 5x3 − 3x.
(a) Show that {f1 , . . . , f4 } is a basis for F3 [x].
(b) If possible, find the unique linear combination of these Legendre polynomials which fit the data
(1,3), (2,4), (3,1), (4,-2). If it is not possible, explain why.
7. Let {1, x, x2 } be a basis for F2 [x], and let {1, x, x2 , x3 , x4 } be a basis for F4 [x]. For each of the following
linear transformations L : F2 [x] 7→ F4 [x], find the matrix representation:
(a) L(p) = xp(x) for any p(x) ∈ F2 [x]
(b) L(p) = x4 + p(x) for any p(x) ∈ F2 [x]
(c) L(p) = x3 p0 (x) − 2x2 p(x) for any p(x) ∈ F2 [x]
8. For each of the linear transformations in Problem 7 find a basis for ker(L) and Ran(L).
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