Math 315 Review Exercise on Definitions - to be done at the end of Chapter 3.

Match each of the following 1-7 to the correct definition chosen from a-f.Some of the definitions may apply to more than one of the numbers 1-7, and at least one of the definitions applies to none of the numbers 1-7.

1. x is a linear combination of  v₁, v₂, ...,  v_n.



2. v₁, v₂, ...,  v_n span the vector space V.



3. Span(v₁, v₂, ...,  v_n) = V.



4. { v₁, v₂, ...,  v_n } is a spanning set for V.



5. v₁, v₂, ...,  v_n are linearly independent.



6. v₁, v₂, ...,  v_n are linearly dependent.



7. { v₁, v₂, ...,  v_n } is a basis of V.

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a. There are scalars c₁, c₂, ...,c_n, not all 0, such that c₁v₁+c₂v₂+...+c_nv_n =0.

b. { v₁, v₂, ..., v_n } is a spanning set for V and is linearly independent.

c. There are scalars c₁, c₂, ...,c_n such that x = c₁v₁+c₂v₂+...+c_nv_n.

d. c₁v₁+c₂v₂+...+c_nv_n =0.

e. For all x in V, there are scalars c₁, c₂, ...,c_n such that x = c₁v₁+c₂v₂+...+c_nv_n.Note:the scalars will be different for different x.

f. If c₁v₁+c₂v₂+...+c_nv_n =0, then c₁= c₂= ...=c_n=0.