Linearly independent set.

Given . Find a vector such that is a linearly independent set.

Response:

Let for arbitrary numbers .

Then for A to be linearly independent, must imply .

Therefore we have the homogeneous linear system, in matrix form:

My question(s):

Do I really have to reduce it to reduced echelon form? It gets really messy!

Is there an easy way to solve this? (Easier than my approach?)

Generalized Cross Product

Cross product - Wikipedia, the free encyclopedia

You can look in the section on generalizations of the cross product into higher dimensions. This will give you a vector that is perpendicular to the subspace spanned by the first 3 vectors. If they are linearly independent, the collection of all 4 should be as well.

Alternatively, guess and check could be popular, you just gotta pick 4 numbers to give you a nonzero determinant that will show your vectors are linearly independent.

Otherwise, yeah, that is probably the best way to do it.