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ORTHOGONALITY

We can use matrix addition and scalar multiplication to combine vectors in a linear combination. The result is a new vector in the same space. For example, in R4, combining three vectors, v, w, and x, and three scalars, s1, s2, and s3, we get y:

(7.6)

Rather than viewing this equation as creating y, we can read the equation in reverse, and imagine decomposing y into a linear combination of other vectors.

A set of n vectors, v1, v2, … , vn, is said to be linearly independent if, and only if, given the scalars c1, c2, … , cn, the solution to the equation:

(7.7)

has only the trivial solution, c1 = c2 = ... = cn = 0. A corollary to this definition is that if a set of vectors is linearly independent, then it is impossible to express any vector in the set as a linear combination of the other vectors in the set.