For each of the situations described below, give an example (if it’s possible) or explain why it’s not possible.

(a) A set of vectors that is linearly dependent. After adding one more vector, the set becomes linearly independent.

(b) A set of vectors in R3 with the following properties (four possibilities):

For each case that is possible, how many vectors could be in the set? (State any constraints, as in “there must be at least…” or “at most…”)

(c) A system of equations with a unique solution. After adding another equation to the system, the new system has infinitely-many solutions.

(d) Vectors v1,v2,v3 ∈R3 that span R3 and satisfy 2v1 + 3v2 + 7v3 = 0

 

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