Linear Independency of $3$ vectors

Question

Consider $3$ vectors of dimension $3$. If these vectors are pairwise linearly independent. Does that mean that those three together are linearly independent? How to prove whether they are or they aren't?

Answer 1

One way to understand this is geometrically. Two vectors in 3 space are independent if they are on distinct lines through the origin. Three vectors in 3 space are independent if you don't lie on a plane through the origin. Taking 3 distinct lines in any plane through the origin gives you your counterexample.

You can also think of this in terms of projective geometry in the plane. Imagine a plane in 3d space not passing through the origin. Then lines in 3 space through the origin (1 dimensional vector spaces) will intersect with this plane to form points and planes through the origin (2-dimensional vector spaces) will intersect to form lines. Two vectors are independent if their projected points are distinct. Three vectors are independent if their projected points are non-collinear. Hence you just take three distinct points along a line to give a counterexample.

Answer 2

Consider these three vectors in $\mathbb R^3$: $(1,0,0)$, $(0,1,0)$, and $(1,1,0)$. Each two of them are linearly independent. What about the set of all three of them?

Answer 3

Not in general, take $(1,0,0)$, $(1,2,0)$ and $(1,3,0)$ they are independent two by two (they form the $xy$ plane). But al three also span the same plane.

To check if they are independent in your case, see if you can write one of them as the linear combination of the other two. You can use roe reduction by forming an strive or finding the determinant or finding the null space or .....