Cauchy Schwarz Inequality

Proposition: Cauchy Schwarz Inequality. Let . Then,

With the equality holding only when , for some

• In other words, the dot product of two vectors will always be less than or equal to the magnitudes of the two vectors
  • The only case where they are equal is when the vectors are linearly dependent, or when the angle between them either 0 or 180.

Remark: implies that , meaning that Therefore, Therefore, we can see that , where is the angle between the two vectors.

• This remark relates to another definition of the dot product
  • We know the dot product can also be defined as: , where is the angle between vectors

Proof: Pick . Consider: This is a quadratic equation in the variable . If is linearly independent, then . This forces the discriminant of the polynomial to be negative, meaning it has no solutions. When is a scalar multiple of , meaning that for some , then has only one root: . This means the discriminant of is zero, so: