Explain why we may assume x10. (Hint: x1 is the first unknown because u₁ was the first vector in our spanning set.) ,un} is a set of vectors that span V, and that {V1, V2, ..., Vk} is an independent set of Suppose {u1, u2, ..., 1 vectors in V. Explain why V1 x1u1 + x2u2+ ...+xnun -must have a solution.

Let V₁, V2, V3 be the vectors in R³ defined by 18 ---D V₁ = -6 V2 = -14 V3 = 20 (a) is (V1, V2, Vs} linearly independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁, V₂, and V3 0 (c) Type the dimension of span {V1, V2, V3}: Note: You can earn partial credit on this problem. -25 -18] 0=v₁+√₂+vs (b) Is (v1, vs} linearly Independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁ and V3. 0=v₁+v₁ V3

Write each vector as a linear combination of the vectors in S. (Use s and s, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) S- ((6, -7, 8, 6), (4, 6, -4, 1)} (a) u- (-54, 127, -128, -72) -(실종 10, ). 41 75 (b) 2 4 (c) (d) z- (4, -2, 3, )

Express the vector of the form v=vi+ v2j + v3k. AB if A is point (-9, -2, 1) and B is point (-4, -9, 4) v - 5i - 7j - 3k v - 51 - 7) + 3k v = 5i + 7] - 3k v - 5i + 7) - 3k

11. a) By solving some equations with some unknowns, determine whether it is possible to write [5, -10, 1] as a linear combination of [4, -8, 4] and [-3, 6, 3], and if it is indeed possible, write the linear combination. 11b) Explain what your result in a) shows about the coplanarity of the 3 vectors.

Use these 3 vectors ?⃗ = (-2, -12, 4), ? = (3, -2, 1), ?⃗ = (8, 12, -4) to verify the the associative property

Can 5 = (7,14,0) be written as a linear combination of vectors from the set {(1,-2,-1), (0,1,2). (3,–1,0)} ? If yes, clearly show how the coefficients are calculated (trial and error will not be accepted). If no, then explain why not. Determine with explanations and calculations whether the set {(-2,3,0). (1, –1,2), (0,3, 2)} is linearly independent.

One of the following vectors lies in span{ 2 + x +x², 3+x – 2x² } (A) 3 + 2x + 4x² (B) 3 + 2x + 5x² (C) 3 + 2x ++ 6x² (D) 3 + 2x + 7x² (E) 3 + 2x + 3x?

Let Is w a linear combination of the vectors V₁, V₂ and 3? choose 13 -14 -4 --[H], -- [5] -- [13] D = = = " " 6 = If possible, write was a linear combination of the vectors ₁, ₂ and 73. For example, the answer w = 4v₁ +50₂ + 603 would be entered 4v1 + 5v2 + 6v3. If cannot be written as a linear combination of the vectors ₁, 2 and 3, enter DNE. V3

Use a counterexample to show that the following statement is false: if a, b and c are nonzero vectors in R^2, then c is a linear combination of a and b.