Exercises 2A 1  Find a list of four distinct vectors in 𝐅3 whose span equals {(𝑥,𝑦,𝑧)∈𝐅3 ∶𝑥+𝑦+𝑧=0}. ^2A-1 2. 2  Prove or give a counterexample: If 𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 spans 𝑉, then the list 𝑣1 − 𝑣2,𝑣2 − 𝑣3,𝑣3 − 𝑣4,𝑣4 also spans 𝑉. ^2A-2 3. 3  Suppose 𝑣1,...,𝑣𝑚 is a list of vectors in 𝑉. For 𝑘 ∈ {1,...,𝑚}, let 𝑤𝑘 = 𝑣1 + ⋯ + 𝑣𝑘. Show that span(𝑣1, ..., 𝑣𝑚) = span(𝑤1, ..., 𝑤𝑚). ^2A-3 4. 4  (a) Show that a list of length one in a vector space is linearly independent if and only if the vector in the list is not 0. (b) Show that a list of length twoinavectorspaceislinearlyindependent if and only if neither of the two vectors in the list is a scalar multiple of the other. ^2A-4 5. 5  Find a number 𝑡 such that (3,1,4),(2,−3,5),(5,9,𝑡) is not linearly independent in 𝐑3. 6. 6  Show that the list (2, 3, 1), (1, −1, 2), (7, 3, 𝑐) is linearly dependent in 𝐅3 if and only if 𝑐 = 8. 7. 7  (a) Show that if we think of 𝐂 as a vector space over 𝐑, then the list 1 + 𝑖, 1 − 𝑖 is linearly independent. (b) Show that if we think of 𝐂 as a vector space over 𝐂, then the list 1 + 𝑖, 1 − 𝑖 is linearly dependent. 8. 8  Suppose 𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 is linearly independent in 𝑉. Prove that the list 𝑣1 − 𝑣2,𝑣2 − 𝑣3,𝑣3 − 𝑣4,𝑣4 is also linearly independent. 9. 9  Prove or give a counterexample: If 𝑣1, 𝑣2, ..., 𝑣𝑚 is a linearly independent list of vectors in 𝑉, then is linearly independent. 10. 10  Prove or give a counterexample: If 𝑣1, 𝑣2, ..., 𝑣𝑚 is a linearly independent list of vectors in 𝑉 and 𝜆 ∈ 𝐅 with 𝜆 ≠ 0, then 𝜆𝑣1, 𝜆𝑣2,..., 𝜆𝑣𝑚 is linearly independent. 11. 11  Prove or give a counterexample: If 𝑣1, ..., 𝑣𝑚 and 𝑤1, ..., 𝑤𝑚 are linearly independent lists of vectors in 𝑉, then the list 𝑣1 + 𝑤1, ..., 𝑣𝑚 + 𝑤𝑚 is linearly independent. 12. 12  Suppose 𝑣1,...,𝑣𝑚 is linearly independent in 𝑉 and 𝑤 ∈ 𝑉. Prove that if 𝑣1 + 𝑤, ..., 𝑣𝑚 + 𝑤 is linearly dependent, then 𝑤 ∈ span(𝑣1, ..., 𝑣𝑚). 13. 13  Suppose 𝑣1, ..., 𝑣𝑚 is linearly independent in 𝑉 and 𝑤 ∈ 𝑉. Show that 𝑣1,...,𝑣𝑚,𝑤islinearlyindependent ⟺ 𝑤∉span(𝑣1,...,𝑣𝑚). 14. 14  Suppose 𝑣1,...,𝑣𝑚 is a list of vectors in 𝑉. For 𝑘 ∈ {1,...,𝑚}, let 𝑤𝑘 = 𝑣1 + ⋯ + 𝑣𝑘. Show that the list 𝑣1, ..., 𝑣𝑚 is linearly independent if and only if the list 𝑤1, ..., 𝑤𝑚 is linearly independent. 15. 15  Explain why there does not exist a list of six polynomials that is linearly independent in 𝒫4(𝐅). 16. 16  Explain why no list of four polynomials spans 𝒫4(𝐅). 17. 17  Prove that 𝑉 is infinite-dimensional if and only if there is a sequence 𝑣1, 𝑣2, ... of vectors in 𝑉 such that 𝑣1, ..., 𝑣𝑚 is linearly independent for every positive integer 𝑚. 18. 18  Prove that 𝐅∞ is infinite-dimensional. 19. 19  Prove that the real vector space of all continuous real-valued functions on the interval [0, 1] is infinite-dimensional. 20. 20  Suppose 𝑝0, 𝑝1, ..., 𝑝𝑚 are polynomials in 𝒫𝑚(𝐅) such that 𝑝𝑘(2) = 0 for each 𝑘 ∈ {0, ..., 𝑚}. Prove that 𝑝0, 𝑝1, ..., 𝑝𝑚 is not linearly independent in 𝒫𝑚(𝐅).