Subspaces of Rn , Bases and Linear Independence Definition. Consider vectors ~v1 , . . . , ~vr in Rn . An equation of the form c1~v1 + · · · + cr ~vr = ~0 is called a linear relation among the vectors ~v1 , . . . , ~vr . If at least one of the ci is nonzero, then we call this a nontrivial linear relation among ~v1 , . . . , ~vr . 2 0 1 7 1 3 1. Let ~v1 = 1, ~v2 = 2, and ~v3 = 4. 5 7 −1 (a) Are there any nontrivial linear relations among these vectors? If so, find one. (b) Are the vectors ~v1 , ~v2 , ~v3 linearly independent? (c) Let V = span(~v1 , ~v2 , ~v3 ). Find a minimal set of vectors that span V . (We’ll call this a basis of V .) How would you describe the shape of V ? 2. Give geometric descriptions of the following: • 1 vector ~v1 in Rn is linearly independent ⇐⇒ . • 2 vectors ~v1 , ~v2 in Rn are linearly independent ⇐⇒ . • 3 vectors ~v1 , ~v2 , ~v3 in Rn are linearly independent ⇐⇒ . • The span of 1 linearly independent vector in Rn is . • The span of 2 linearly independent vectors in Rn is . • The span of 3 linearly independent vectors in Rn is . • The smallest span of vectors is . 1 x x 2 3. Let V = ∈ R : x ≥ 0 . (In words, V is the set of vectors in R2 with x ≥ 0.) Is V a subspace y y of R2 ? Why or why not? x 2 4. Let V = ∈ R : y = ±x . Is V a subspace of R2 ? Why or why not? y x 2 5. Let V = ∈ R : y = 3x + 1 . Is V a subspace of R2 ? Why or why not? y 2 6. Decide whether each of the following planes in R3 is a subspace of R3 . If so, find a basis of the subspace. x (a) The plane consisting of all vectors y satisfying 2x − 7y + 4z = 0 z x (b) The plane consisting of all vectors y satisfying 2x − 7y + 4z = 1 z 3 7. Let A be an n × m matrix. Is im A a subspace of Rn ? Why or why not? 8. True or false: If ~v1 , . . . , ~v5 are linearly dependent vectors in R7 , then ~v5 must be in span(~v1 , . . . , ~v4 ). 9. If ~v1 , . . . , ~vr are vectors in Rm , show that span(~v1 , . . . , ~vr ) is closed under addition and scalar multiplication. 4
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