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Question
-For parallel we have larned that to vectors are parallel if there exists a lambda in R so that vector v = (lambda)(vector w) OR vector w = (lambda)(vector v)

I dont understand what to do for this probem at all. Thanks for any help.

For any two vectors vector v, vector w in R^n.

(a) State the de nition of linearly dependence and the definition of parallel for {vector v vector w}.
(b) Assume that vector v and vector w are linearly dependent. Show that they are parallel using the definitions of (a).
(c) Assume that vector v and vector w are parallel. Show that they are linearly dependent using the definitions of (a).

So far, for part a) I have obtained:
- For linear dependence: Let vector v and vector w be in R^n.
vector v and vector w are linearly dependent if there are constants not all zereos so that a_1(vector v)+a_2(vector w)= vector 0.

For part b I was thinking of maybe plugging in the value of v from the definition of parallel into that of linear dependence and say that the vectors are still parallel because they are just being stretched due to being multiplied by constants a and lambda. For part c i have no clue. Thanks for the help on this problem.

(a) They are linearly dependent if there is some constant C such that Cv = w.
They are parallel if their cross product, v x w, is 0.

(b) If they are linearly dependent, then Cv = w.
We can take v x w and change it to v x (Cv) = C(v x v).
The cross product of a vector with itself is 0, and times a constant is still 0.

(c) If the vectors are parallel, then we have Cv = w.
That is the definition of linearly dependent.

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