ASSIGNMENT 2–MATH 2100 Due Monday, September 27 at 10:35 ...

ASSIGNMENT 2–MATH 2100
Due Monday, September 27 at 10:35 in tutorial.
Suppose V and W are vector spaces over a field F .
1. Suppose ϕ : V → W is an isomorphism. Recall that the inverse func- /10
tion ϕ −1 : W → V is defined in the following way. For any w ∈ W there
exists some v ∈ V such that ϕ(v) = w (surjectivity of ϕ). Moreover,
this v is unique (injectivity of ϕ). Therefore it makes sense to define
ϕ −1 (w) = v. This has the consequences that
and
ϕ(ϕ −1 (w)) = ϕ(v) = w
ϕ −1 (ϕ(v)) = ϕ −1 (w) = v.
Prove that ϕ −1 : W → V is an isomorphism, by verifying that it is
linear and bijective.
2. Suppose x, y, z ∈ V satisfy x + y + z = 0. Prove that span{x, y} =
span{y, z}. /10
3. Recall that F [x] is the vector space (over F ) of polynomials with coefficients
in x, i.e.
F [x] = {a0 + a1x + · · · + anx n : a0, . . . an ∈ F, n ≥ 0}.
Fix a positive integer m. For this exercise you may use without proof
that the subset W ⊂ F [x] of all polynomials of degree m or less is a
subspace of F [x] and that this subspace has two ordered bases
and
B = {1, x, x 2 , . . . , x m }
B ′ = {1, x, x 2 , . . . , x m−1 , x m−1 + x m }.
(a) Determine the basechange matrix P ∈ F m+1×m+1 such that B ′ =
BP and compute the determinant of P . (Hint: Determine P by
specifying its matrix entries pij ∈ F . Show your work!)
/10

(b) Suppose here that F = R and {f1, . . . , fm+1} ⊂ W such that /10
fj(1) = 0
for all 1 ≤ j ≤ m + 1. Prove that {f1, . . . , fm+1} is linearly
dependent. (Hint: What is the dimension of W ?)
SUGGESTED EXERCISES
Chapter Exercises
3 3.1, 3.4, 3.5, 3.8., 4.1-4.3