Ok. So is it about trying to show two vectors are linearly independent and they clearly are like i & j ? ( non-zero determinant?)

Linear algebra is overkill for projectile problems.

Remember that there is no requirement for basis vectors to be orthogonal.

They just can't be parallel.

If they are EDIT orthogonal /EDIT <parallel> the set of unit vectors is called an orthonormal basis.

This is usually the most useful to work with, but not always.

However we know by definition that horizontal and vertical are orthogonal directions.

So any vector pointing vertically will be orthogonal to any vector pointing horizontally.

Now possible vectors pointing vertically (horizontally)

Velocity

Momentum

Acceleration

And that which is often forgotten

The unit (vertical) vector.

Now if we multiply unit vector by a scalar coefficient we get a vector pointing

In the direction of the unit vector.

In this way we can make distance into a vector.

Of course we can also regard velocity etc as coefficient multiples of the one unit vector.

Does this help?