Respuesta :
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).
[tex]\left|\begin{matrix}-1 & 5 & 3 \\ 4 & 2 & -5 & \\ 6 & -3 & -4\end{matrix}\right| = -119\neq 0[/tex]
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 [tex]\mathbb{R}^3[/tex] 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 [tex]\mathbb{R}^3[/tex], 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
