Vectors as Functions

Describing Vectors as Functions

• A vector doesn't have a clear definition
• As long as there's a reasonable notion of scaling and adding, a vector can be any object

• The following are just a few examples of vectors:

  • An arrow on a flat plane that we can describe with coordinates
  • A pair of real number that is just nicely visualized as an arrow on a flat plane
  • The set of all $\pi$ creatures
  • A set of any other crazy object
• Again, the above are examples of vectors as long as they have a reasonable notion of scaling and adding
• However, possibly the best way to think about vectors is as functions

Motivating Vectors as Functions

• A vector can be thought of as a type of function
• Similar to how the only thing vectors can really do is be added together or scaled by a real number, we can only add functions together and scale them by real numbers

• The following are some examples of this idea:

  • Adding two functions $f$ and $g$ provides us with a new function $f+g$
  • Scaling a function $f$ by a real number $2$ will give us a new function $2f$
• Similar to how we can use linear transformations to map a vector from one vector space to another vector space, we can use operators to map one function space to another function space

• One familiar example of this is the derivative

  • It's an operator that transforms one function space into another function space

Linear Transformations as Operators

• Similar to linear transformations needing to be linear, operators need to maintain linearity as well

• A transformation is linear if it satisfies two properties

  • Additivity
  • Scaling
• Additivity means that if you add two vectors $v$ and $w$ together, then apply a transformation to their sum, you get the same result as if you added the transformed versions of $v$ and $w$
• The scaling property is that when you scale a vector $v$ by some number, then apply the transformation, you get the same vector as if you scale the transformed version of $v$ by that same amount
• In other words, linear transformations need to preserve the operations of vector addition and scalar multiplication
• The idea of gridlines remaining parallel and evenly spaced is just an illustration of what these two properties mean in a 2D space
• The properties of additivity and scaling need to be maintained for functions (or operators) too
• For example, if you add two functions, then take the derivative, it's the same as first taking the derivative of each function separately, then adding the result
• Similarly, if you scale a function, then take the derivative, it's the same as first taking the derivative, then scaling the result

References

• Essence of Linear Algebra Video
• Properties of Differential Operators Wiki