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Find the value(s) of $h$ for which the vectors are linearly dependent. Justify each answer.$\left[\begin{array}{r}{1} \\ {5} \\ {-3}\end{array}\right],\left[\begin{array}{r}{-2} \\ {-9} \\ {6}\end{array}\right],\left[\begin{array}{r}{3} \\ {h} \\ {-9}\end{array}\right]$

The vectors are linearly dependent for all $h$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

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for this exercise. We are provided with three different vectors, and the vector V three we see here also depends on the value of age. So questions we might ask about this particular set of vectors are for what values of age would this set be linearly dependent? Well, to answer that kind of question, we can start by making a Matrix A, which is formed using the columns V. One, V two and V three are provided vectors. Then our metric say would be 15 negative three for the first column. Negative to negative 96 and call him, too, and three h negative nine for the third column. Let's re reduce this toe echelon form. That way we can analyze the pivots and now determined linear dependence. So let's begin then with Row one or entry here our first pivot position and will take out the five and the negative three below. So let me copy one negative two and three for a first step. Multiply row one by negative five. Added to row two, we obtained 01 and H minus 15 for the next step. If we triple row one multiplied by three and after Row three. Then we'll produce a zero zero and zero as well. So that's kind of convenient. With one row operation, we are in echelon form now. The pivots will be here and here and nowhere else. And the value of H does not change where the pivots will be. Let's consider that matrix equation eight times X equals a zero vector. This equation has nontrivial solutions. The reason we know this is that there is a free variable that corresponds to the third column in this particular matrix. So since we have non trivial solutions, we know nontrivial solutions to a homogeneous matrix equation implies immediately that the set of vectors V one, V two and V three is linearly dependent, and the linear dependence we found again does not depend on H whatsoever. So V one v two V three is linearly dependent for all values h in the set of Real's. So this is how we can determine whether or not we have linear, independent or dependent vectors in this situation,

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