A subset S of vector space V is called linearly dependent if there exist a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that
Note that the zero on the right is the zero vector, not the number zero.
Here your distinct vectors are x, y, z as opposed to v_1, V_2,...,v_n. since you are considering only 3 vectors, you can prove they are linearly independent by showing there are distinct constants (that's why it says not all zero) that you can multiply each vector by and get the zero vector. this proof simply chose those three constants to be 1, -k, and 0.
(1)y + (-k)x + (0)z = 0
You get all the other steps right?