MATH 110: LINEAR ALGEBRA MIDTERM #1 REVIEW

Problem 1. Let V = \{a + b\sqrt{2} : a, b ∈ ℚ\}. (a) ℝ is a vector space over the rational numbers ℚ. Prove that V is a subspace of ℝ (over ℚ).

(b) Show that β = \{1, \sqrt{2}\} is a basis for V.

(c) Prove that the map T : V → V by x ↦ x\sqrt{2} is a linear transformation.

(d) Compute [T]_{β}.

(e) Prove that T is an isomorphism.

Problem 2. Let V be a finite-dimensional vector space, and let W be a subspace of V. Show that there exists a subspace Z ⊂ V such that V = W ⊕ Z.

Problem 3. Let V and W be finite-dimensional vector spaces over a field F, and let

V × W = {(v, w) : v ∈ V, w ∈ W }.

Then V × W is a vector space over F, by

(v_{1}, w_{1}) + (v_{2}, w_{2}) = (v_{1} + v_{2}, w_{1} + w_{2}) and c(v, w) = (cv, cw)

for v_{1}, v_{2} ∈ V, w_{1}, w_{2} ∈ W, and c ∈ F.

(a) What is the dimension of V × W?

(b) Prove or disprove: V × W is isomorphic to L(V, W).

Problem 4. Let T : P_{2}(ℝ) → P_{2}(ℝ) be the linear transformation defined by f(x) ↦ f′(x). Let β be the ordered basis 1, x, x^{2} of P_{2}(ℝ), and let β′ be the ordered basis 1, 1 + x, 1 + x + x^{2}. Find a matrix Q such that

[T]_{β′} = Q^{−1}[T]_{β}Q.

Problem 5. Let v_{1} = (2, 1) and v_{2} = (1, −1). Then β = v_{1}, v_{2} is an ordered basis for V = ℝ^{2}. (a) Prove that there exist linear functionals f_{1}, f_{2} : V → ℝ satisfying:

f_{1}(v_{1}) = 1, f_{1}(v_{2}) = 0; f_{2}(v_{1}) = 0, f_{2}(v_{2}) = 1.

(b) Find a formula for f_{1}(x, y).

(c) Prove that {f_{1}, f_{2}} is a basis for V^{∗}.