Find the value(s) of h for which the vectors are linearly dependent. Justify each answer. [15−3],[−2−96],[3h−9]

Question

Find the value(s) of h for which the vectors are linearly dependent. Justify each answer.  [15−3],[−2−96],[3h−9]

un4t5o4v · Accepted Answer

Step 1  Definition: A set of vectors  {v1,…..,vn} is linearly dependent  x1=x2=…=xn=0  is NOT the only solution to the equation  x1v1+x2v2+…+xnvn=0     (1)  Of course  x1=x2=…=xn=0  will always be one solution to this equation, but it may or may not be the only solution.  Step 2  Notice that in this case, equation (1) is equivalent to the equation  Ax=0  A=[1−235−9h−36−9]  x=[x1x2x3]  Step 3  We can find the solution to  Ax=0  by row reduction:  [1−23|05−9h|0−36−9|0]∼[1−23|001h−15|0000|0]  (R2→R2−5R1R3→R1+2R2)  ∼[10−27+2h|001h−15|0000|0]      (R1→R1→+2R2)  Step 4  From here we see that for any value of h, the possible solutions of Ax=0 are  [x1x2x3]=[(27−2h)x3(15−h)x3x3]=x3[27−2h15−h1]  Since the last coordinate can be anything and is independent of h , we see that regardless of the value of h, this set of vectors is always linearly dependent.  Result:  The vectors are linearly dependent for all h

karton · Accepted Answer

Step 1:Let&#039;s set up the following equation:c1[153]+c2[−2−96]+c3[3h−9]=[000]Expanding this equation, we get the following system of equations:c1−2c2+3c3=05c1−9c2+hc3=03c1+6c2−9c3=0We can solve this system of equations to find the values of h. First, let&#039;s eliminate c1 from the second and third equations by subtracting the first equation multiplied by 5 from the second equation and by subtracting the first equation multiplied by 3 from the third equation:−9c2+5c3=06c2−9c3=0Step 2:Now, let&#039;s express c2 and c3 in terms of t as follows:c2=59c3c3=tSubstituting these expressions into the equations, we get:−9(59t)+5t=06(59t)−9t=0Simplifying, we have:−5t+5t=0109t−9t=0The first equation is satisfied for any value of t. For the second equation, we can find the value of t that makes it true:109t−9t=0109t=9t109=910=9×910=81The equation 10=81 is not true for any value of t. Therefore, there is no value of t that satisfies the second equation.Since the second equation cannot be satisfied, there are no values of h that make the given vectors linearly dependent.

user_27qwe · Accepted Answer

Answer:There are no values of h that make the given vectors linearly dependent.Explanation:To find the value(s) of h for which the vectors are linearly dependent, we need to determine when one of the vectors can be written as a linear combination of the other two vectors.Let&#039;s set up the equation [153]=a[−2−96]+b[3h−9], where a and b are scalars.Expanding the equation, we get the following system of equations:−2a+3b=1   (1)−9a+hb=5   (2)6a−9b=3    (3)To determine when the vectors are linearly dependent, we need to find values of h that satisfy this system of equations.Let&#039;s solve this system of equations. By manipulating (1) and (3), we can eliminate b:−2a+3b=1    (1)6a−9b=3    (3)Multiplying (1) by 3 and (3) by 1, we obtain:−6a+9b=3    (4)6a−9b=3    (3)Adding (4) and (3), we eliminate b:0a+0b=6This equation has no variables, and the left side is nonzero. Therefore, there is no solution for h that makes the vectors linearly dependent.

alenahelenash · Accepted Answer

To determine the values of h for which the given vectors are linearly dependent, we need to find if there exist non-zero constants c1, c2, and c3 such that the equationc1[153]+c2[−2−96]+c3[3h−9]=[000] has a non-trivial solution.We can rewrite this equation as a system of linear equations:c1−2c2+3c3=05c1−9c2+hc3=03c1+6c2−9c3=0To solve this system, we can use row reduction on the augmented matrix:[1−2305−9h036−90]Applying row operations, we can simplify the matrix:[1−23001−h500000]From the row-reduced form, we can see that the system has infinitely many solutions (linearly dependent) if and only if the last row is all zeros. In this case, we have:0x1+0x2+0x3=0This equation is always satisfied, regardless of the value of h. Therefore, for any value of h, the given vectors are linearly dependent.To summarize, the vectors are linearly dependent for all values of h.

Find the value(s) of h for which the vectors are linearly dependent. Justify eac

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