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Math 2R03 (Theory of Linear Algebra) Homework Assignment 2
All of the questions from Part A will be graded. One of the questions from Part B will be graded in detail, while the other will be marked for completion. Assignments will be submitted via Crowdmark. You will be graded on your solution and how well you write your proof.
Part A. [Short Questions; 4pts]
Exercise 1. Suppose that V = span(v1, v2, v3). Prove that the list v1 + v2 + v3, v2, 2022v3 also spans V .
Exercise 2. Let V = C 2 . Note that V is a vector space over R if we use the scalar operation r(a + bi, c + di) = (ra + (rb)i, rc + (rd)i)
for any r ∈ R and (a + bi, c + di) ∈ C 2 . At the same time, V is also a vector space over C if we use the scalar operation z(a + bi, c + di) = (z(a + bi), z(c + di))
for any z ∈ C and (a + bi, c + di) ∈ C 2 . Show that if we view V as a vector space over C, then the vectors (1, i),(i, −1) ∈ V are linearly dependent, but if we view V as a vector space over R, then (1, i),(i, −1) ∈ V are linearly independent.
Part B. [Proof Questions; 6pts]
Exercise 3. Let p1(x) = 3x + x 3 and p2(x) = 2022 be elements in the vector space V = P3(R). Extend {p1(x), p2(x)} to a basis of V .
Exercise 4. Prove that the functions sin x,sin 2x, and sin 3x are linearly independent on the interval [0, π].
Hint. Assume c1 sin x + c2 sin 2x + c3 sin 3x = 0. To show that ci = 0, multiply through by sin ix, and integrate over the interval [0, π]. The following identity will also be useful: sin A sin B = 1 2 [cos (A − B) − cos (A + B)].
Remark. The above result can be extended to show that sin x,sin 2x, . . . ,sin mx are linearly independent on [0, π] for any integer m ≥ 1. The fact that these functions are linearly independent plays an important role in Fourier Series.
Additional Suggested Problems. [Not graded]
Problems 1.C # 19, 20, 2.A # 8, 9, 10