To prove vector independence, you must show that no linear combination of the given vectors can equal the zero vector, except for the trivial solution where all coefficients are zero.
Two vectors are considered not parallel if they do not have the same direction or are not multiples of each other. In other words, they do not lie on the same line.
Proving vector independence is important in linear algebra and other areas of mathematics because it allows us to determine if a set of vectors can form a basis for a vector space. It also helps us understand the relationships between vectors and their linear combinations.
A linear combination of vectors is the sum of scalar multiples of those vectors. For example, if you have vectors v and w, their linear combination can be written as av + bw, where a and b are scalars.
Yes, for two vectors a = [1, 2, 3] and b = [4, 5, 6], we can show that they are independent by setting up the equation ax + by = 0 and solving for x and y. If the only solution is x = 0 and y = 0, then the vectors are independent.