Orthogonalizing vectors. Suppose that a and b are any n-vectors. Show that we can always find a scalar γ so that (a − γb) ⊥ b, and that γ is unique if b 6= 0. (Give a formula for the scalar γ.) Roughly speaking, we can always subtract a multiple of a vector from another one, so that the result is orthogonal to the original vector. This is called orthogonalization, and is a basic idea used in the Gram-Schmidt algorithm.

Step-by-step explanation:

The question to be solved is the following :

Suppose that a and b are any n-vectors. Show that we can always find a scalar γ so that (a − γb) ⊥ b, and that γ is unique if . Recall that given two vectors a,b  a⊥ b if and only if where is the dot product defined in . Suposse that . We want to find γ such that . Given that the dot product can be distributed and that it is linear, the following equation is obtained

Recall that are both real numbers, so by solving the value of γ, we get that

By construction, this γ is unique if , since if there was a such that , then