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Engineering Calculus Notes 44

# Engineering Calculus Notes 44 - 1.13 is the trivial one(a...

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32 CHAPTER 1. COORDINATES AND VECTORS 5. Show (using vector methods) that the line segment joining the midpoints of two sides of a triangle is parallel to and has half the length of the third side. 6. Given a collection { −→ v 1 , −→ v 2 ,..., −→ v k } of vectors, consider the equation (in the unknown coefficients c 1 ,. . . , c k ) c 1 −→ v 1 + c 2 −→ v 2 + ··· + c k −→ v k = −→ 0 ; (1.13) that is, an expression for the zero vector as a linear combination of the given vectors. Of course, regardless of the vectors −→ v i , one solution of this is c 1 = c 2 = ··· = 0; the combination coming from this solution is called the trivial combination of the given vectors. The collection { −→ v 1 , −→ v 2 ,..., −→ v k } is linearly dependent if there exists some nontrivial combination of these vectors—that is, a solution of Equation ( 1.13 ) with at least one nonzero coefficient. It is linearly independent if it is not linearly dependent—that is, if the only solution of Equation (
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Unformatted text preview: 1.13 ) is the trivial one. (a) Show that any collection of vectors which includes the zero vector is linearly dependent. (b) Show that a collection of two nonzero vectors { −→ v 1 , −→ v 2 } in R 3 is linearly independent precisely if (in standard position) they point along non-parallel lines. (c) Show that a collection of three position vectors in R 3 is linearly dependent precisely if at least one of them can be expressed as a linear combination of the other two. (d) Show that a collection of three position vectors in R 3 is linearly independent precisely if the corresponding points determine a plane in space that does not pass through the origin. (e) Show that any collection of four or more vectors in R 3 is linearly dependent . ( Hint: Use either part (a) of this problem or part (b) of Exercise 4 .)...
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