Orthogonality and linear independence

Let and be vectors in a vector space equipped with an inner product (in the scope of MAT 224, the only examples are subspaces of ).

Proposition

Proof

To show linear independence we have to show that any linear combination of and that is equal to has to have all coefficients to be . Let’s take such a linear combination: .

How to use orthogonality? We have an equality of vectors, and we can apply the same operation to both sides to get a new equality. Let’s apply the dot product with the vector to both sides:

At the same time,

We get that . From properties of the dot product only if . Therefore, has to be satisfied as we are given .

Repeating the same reasoning with applying the dot product with gives us that . Therefore, the linear combination has to be trivial, and and are linearly independent.

Corollary

Given a set of mutually orthogonal vectors in (that is, if ). Then are linearly independent.