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QUESTION: 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 denition 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).

ANSWER: Billy, Please send me the definitions of linear dependence and parallel that you have been taught and I'll take a whack at parts a and b.

---------- FOLLOW-UP ----------

QUESTION: Sorry for the delay on this.

-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)

- 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.

Thanks for any further help with this problem. 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.

With these definitions, b) and c) are pretty easy.

For b), your approach is correct: linear dependence ⇒ av + bw = where a,b ≠ 0 so that v = -(b/a)w which is the definition of parallel (important that a ≠ 0)

c) parallel ⇒ v = λw or v - λw = 0 where the coefficients of v and w are non-zero ⇒ linearly independent.

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