Let and be any three non-zero vectors. Then, is independent ofx

To prove that the vector is independent of , we need to show that no linear combination of and can result in the zero vector, unless is zero.

Step 1: Assume that and are non-zero vectors.

Step 2: Consider a linear combination of and , given by:

= +

Step 3: We want to show that the only solution to this equation is when is zero.

Step 4: Assume that is not zero. This means that at least one of the coefficients or is non-zero.

Step 5: Without loss of generality, let's assume that is non-zero.

Step 6: Now, we can rewrite the equation as:

= +

Step 7: Rearranging the equation, we have:

= -

Step 8: Dividing both sides of the equation by , we get:

= -

Step 9: Since is non-zero, we can divide both sides of the equation by , giving us:

= -

Step 10: This implies that is a linear combination of and , which contradicts the assumption that and are independent.

Step 11: Therefore, our initial assumption that is not zero must be incorrect.

Step 12: Hence, we can conclude that the vector is independent of .