Which of the following sets of vectors are linearly independent?

Question

A) [8,1,-6], [-1,-5,-4]
B) [9,3,0], [-7,5,0], [-8,6,0]
C) [-1,-7],[9,-1]
D) [2,4,7],[-9,-2,-3],[7,-2,-4]
E) [7,-9], [-2,-3], [-4,-6]
F) [-4,-6], [4,6]

Answer 1

Dalia,

 

The concept of linear independence is really important, and I found difficult when I first encountered it. But, basically, a set of vectors are independent if one of them cannot be created from the others in the set, through addition and scaling. The importance of the concept is that is has to do with how many vectors are needed to "span the vector space."

 

So, in the first example A) these are independent because there are 2 vectors in 3 space (xyz - space), pointing in different directions - one cannot be created from the other - they point in different directions and together define a plane in 3 space. How do we know they are pointing in different directions? You can use the dot product to find the angle between them, or just show that the rations of the x, y, and z components are such that one cannot be created from the other. Use whatever techniques you were given in the lecture or book.

 

For B, you can tell by inspection since they all 3 have a z component of 0, then they are all 3 in the xy plane, so they must be dependent. Now, for C, these are 2 vectors in 2-space (only 2 components), and they point in different directions. For D, you can take the determinant of the 3x3 matrix composed of these 9 numbers and show it is non zero, which means they are independent. You should use the techniques that were taught in class to demonstrate. For E, you have 3 vectors in 2 space, so there are more vectors than needed to span 2 dimension - they must be dependent. Show that you can crate one from the other. And for F, you can see that one of the vectors is -1 times the other - so use that information.

 

- Ben