## For what values of h are the vectors [Start 3 By 1 Matrix 1st Row 1st Column negative 1 2nd Row 1st Column 4 3rd Row 1st Column 6 EndMatrix

Question

For what values of h are the vectors [Start 3 By 1 Matrix 1st Row 1st Column negative 1 2nd Row 1st Column 4 3rd Row 1st Column 6 EndMatrix ], [Start 3 By 1 Matrix 1st Row 1st Column 5 2nd Row 1st Column 2 3rd Row 1st Column negative 3 EndMatrix ], [Start 3 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column negative 5 3rd Row 1st Column negative 4 EndMatrix ], and [Start 3 By 1 Matrix 1st Row 1st Column 12 2nd Row 1st Column negative 20 3rd Row 1st Column h EndMatrix ]linearly dependent?

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2021-09-15T03:21:41+00:00
2021-09-15T03:21:41+00:00 1 Answer
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## Answers ( )

Answer:The set is linearly dependent for every value of h.

Step-by-step explanation:Recall that a set of vector {a,b,c,d} is a linearly dependent set of vectors if any of the vectors can be written as a linear combination of the other ones.

We are given the following vectors.a=(-1,4,6), b=(5,2,-3), c=(3,-5,-4), d=(12,-20, h). At first, we will check if a,b,c are linearly independent. To do so, we will calculate the following determinant (the procedure of the calculation is omitted).

Since the determinant is not zero, this implies that the vectors a,b,c are all linearly independent. Since a,b,c are all vectors in which is a 3-dimensional space, and they are 3 linear independent vectors, then they are automatically a base of this space. Consider now the vector d. Since {a,b,c} is a base of , then it generates any vector of this space(i.e any other vector of the space is a linear combination of {a,b,c}). In particular, d. So the set {a,b,c,d} is linearly independent for any value of h