Transforming vector-valued-functions
The difference between a function and a transformation
If you’ve taken Algebra or Calculus, you’re familiar with the idea of a function, which is a rule that maps one value to another.
For instance, the function ???f(x)=x+1??? maps ???x??? to ???x+1???. It tells us that, if we put any value ???x??? into the function ???f???, the function will give back ???x+1???. In other words, the function will always return an output value that’s related to the input value we gave it.
We can also write the function ???f(x)=x+1??? as ???f:\ x\mapsto x+1???, where the arrow with the line on the back literally means “maps to,” telling us that ???f??? will map every ???x??? to ???x+1???.
Functions vs. transformations
We can also use functions to map vectors. For instance, the function
tells us that, for every vector ???\vec{v}=(v_1,v_2)??? that we put into ???f???, the function will give us back a new vector, ???\vec{v}=(3v_1-v_2,-2v_2)???. When a function maps vectors, we call it a vector-valued function.
While we usually use functions to map coordinate points, if we’re going to map vectors from one space to another, we usually switch over from the language of “functions,” to “transformations.”
In other words, even though functions and transformations perform the same kind of mapping operation, if we want to map vectors, we should really say that the mapping is done by a transformation instead of by a function. So instead of writing
to express the transformation of a vector ???\vec{v}=(v_1,v_2)???, it’s more appropriate to write
In the same way that it’s most common to use ???f??? to indicate a function, it’s most common to use ???T??? to represent a transformation.
Domain, codomain, and range
Where before we used the notation ???f:\ x\mapsto x+1??? to describe the mapping done by the function, we can use a regular arrow like ???T:\ A\to B??? to indicate that the transformation ???T??? is mapping vectors from the set (or space) ???A??? onto vectors in the set (or space) ???B???.
We also want to always consider the space of what we’re mapping from and what we’re mapping to. For instance, with ???T:\ A\to B???, let’s say we’re mapping from real numbers to real numbers, where both vector sets ???A??? and ???B??? are defined by real numbers. We could write
???T:\ \mathbb{R}\to\mathbb{R}???
More specifically, if ???T??? is mapping from the two-dimensional real plane to the two-dimensional real plane, we could write
???T:\ \mathbb{R}^2\to\mathbb{R}^2???
Keep in mind that, in Linear Algebra, we’ll sometimes be mapping “across dimensions,” for instance, from two dimensions to three dimensions, or vice versa.
In any transformation, the domain is what we’re mapping from, and the codomain is what we’re mapping to. So if ???T:\ \mathbb{R}^2\to\mathbb{R}^3???, then the domain would be the two-dimensional plane ???\mathbb{R}^2???, and the codomain would be three-dimensional space ???\mathbb{R}^3???. On the other hand, if ???T:\ \mathbb{R}^3\to\mathbb{R}^2???, then the domain would be ???\mathbb{R}^3??? and the codomain would be ???\mathbb{R}^2???.
The range is within the codomain. It’s the specific set of points that the mapping actually maps to inside the codomain. In other words, ???T??? might be mapping us into ???\mathbb{R}^3??? in general, but ???T??? might not be mapping to every single point in ???\mathbb{R}^3???. Whatever set of vectors in ???\mathbb{R}^3??? are actually getting mapped to will make up the range of the ???T???.
Transformations are vector-valued functions
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Finding the domain, codomain, and range of a transformation
Example
The transformation ???T??? maps every vector in ???\mathbb{R}^4??? to the zero vector ???\vec{v}=(0,0)??? in ???\mathbb{R}^2???. What are the domain, codomain, and range of ???T????
Because ???T??? is mapping vectors in ???\mathbb{R}^4??? to vectors in ???\mathbb{R}^2???, we can express ???T??? as ???T:\ \mathbb{R}^4\to\mathbb{R}^2???, and say that the domain of the transformation is ???\mathbb{R}^4??? and its codomain is ???\mathbb{R}^2???.
If every vector in ???\mathbb{R}^2??? was being mapped to by ???T???, we would say that the range of ???T??? is ???\mathbb{R}^2???. But the transformation is mapping every vector in ???\mathbb{R}^4??? to only the zero vector ???\vec{v}=(0,0)??? in ???\mathbb{R}^2???. Therefore, the range of ???T??? is just the zero vector, ???\vec{v}=(0,0)???.