Does the concept of linear independence apply to polynomials of more than one va

Question

Does the concept of linear independence apply to polynomials of more than one variable? If so, how does one go about determining whether said polynomials are linearly independent? For example, I am considering the polynomials x3 - 3xy2, y3 - 3yz2, and 5z3 - 3z(x2+y2+z2). I am interested in a direct mathematical way to show this. I understand that linear independence implies that the coefficients of a linear combination of, in this case, polynomials must be zero to imply that the combination is zero.

Explanation / Answer

YOU CAN APPLY SAME METHODS AS FOR SINGLE VARIABLE..AS FOLLOWS..

1. FIRST TREAT IT AS A FUNCTION OF X ONLY ..FIND OUT THE CONDITION NEEDED FOR THEM TO BE LINEARLY DEPENDENT OR INDEPENDENT ...

NATURALLY THE GOVERNING CONDITIONS WILL INVOLVE FUNCTIONS OF Y & Z..

TREAT THEM AS CONSTANTS ..

2. NOW TREAT IT AS A FUNCTION OF Y ONLY ..FIND OUT THE CONDITION NEEDED FOR THEM TO BE LINEARLY DEPENDENT OR INDEPENDENT ...

NATURALLY THE GOVERNING CONDITIONS WILL INVOLVE FUNCTIONS OF X AND Z

TREAT THEM AS CONSTANTS ..

3. NOW  TREAT IT AS A FUNCTION OF Z ONLY ..FIND OUT THE CONDITION NEEDED FOR THEM TO BE LINEARLY DEPENDENT OR INDEPENDENT ...

NATURALLY THE GOVERNING CONDITIONS WILL INVOLVE FUNCTIONS OF X AND Y

TREAT THEM AS CONSTANTS ..

IF IT PASSES THE TEST OF L.D. OR L.I. IN EACH CASE THEN AND THEN ONLY HEY ARE L.I. OR L.D.

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