Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
Math 2R03 Theory of Linear Algebra Homework Assignment 1
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. Let V be a vector space over a field F. Show that for all x ∈ V there is a unique y such that x + 2022y = 0.
Exercise 2. Suppose that U and W are subspaces of a vector space V . Prove that if W ⊆ U, then U + W = U.
Part B. [Proof Questions; 6pts]
Exercise 3. Let V = F ∞ and let
W = {(x1, . . . , x2021, 0, x2023, x2024, . . .) | xi ∈ F} ⊆ V.
That is, W consists of all the elements of F ∞ whose 2022nd coordinate is zero. Prove that W is a subspace of V .
Exercise 4. Consider the subspace U = {(x, y, 2022x) | x, y ∈ R} ⊆ R 3 . Find a subspace W ⊆ R 3 such that R 3 = U ⊕ W.
Hint. Make sure you prove that your set W is a subspace and U + W = R 3 .
Additional Suggested Problems. [Not graded]
Problems 1.A # 11, 1.B # 1, 2, 1.C # 6, 10 1